About this topic
How many ways can you do something? That sounds tricky. It is not.
The big secret is one little word: order. List one way. Then the next. Then the next. Go slow.
When you go in order, two good things happen. You never miss a way. And you never count the same way twice.
In this lesson you will list, draw little trees, make pairs, count pictures, and learn a clever shortcut. All with small numbers. Let's go!
List them in order
Two scoops of ice cream. You can pick cherry, lemon, or mint. Two different flavors. How many ways?
You could guess. But guessing makes you miss some. Instead, do the calm thing: go in order. Start with cherry. Use it up first. Then move to lemon.
Cherry first, all of it. Then lemon. Mint never has to start — it already got matched.
Three ways. See how tidy that is? You go down the list and you can tell, with your eyes, that none are missing.
After cherry+lemon, do not add lemon+cherry. Same two scoops! Going in order saves you from counting it twice.
Pick an order and stick to it. Smallest first, or left to right. Then go slow and write each way down as you find it.
Do not write the same pair backwards. cherry+lemon and lemon+cherry are the same two scoops. Count it once.
The numbers 1, 2, 3, 4 and 5 are written on the board. Ali chooses 2 of them and adds them together. How many different results are possible?
The numbers 1, 2, 3, 4, 5 are on the board. Ali picks 2 and adds them. How many different answers can he get?
You do not have to try every pair. Find the smallest answer and the biggest first. Smallest: 1 + 2 = 3. Biggest: 4 + 5 = 9.
Now walk up from 3 to 9 and check you can make each one. 3 (1+2), 4 (1+3), 5 (1+4), 6 (2+4), 7 (3+4), 8 (3+5), 9 (4+5). Every one works!
Count them: 3, 4, 5, 6, 7, 8, 9. That is 7 different answers.
Find the smallest and the biggest first. Then walk up the numbers in between and check each one. Count the answers you can make.
Go in order, smallest first. List slow. Never write the same way twice.
2018 · #12 You make two-digit numbers using the digits 2, 0, 1 and 8. Each number must be bigger than 10 and smaller than 25, and made of two...
You make two-digit numbers using the digits 2, 0, 1 and 8. Each number must be bigger than 10 and smaller than 25, and made of two different digits. How many different numbers do you get?
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- The number is bigger than 10 and smaller than 25, so it starts with 1 or 2.
- Starting with 1, the other digit (from 2, 0, 8) gives 12, 10, 18 — but 10 is not bigger than 10, so keep 12 and 18.
- Starting with 2, staying under 25, gives 20 and 21. All together: 12, 18, 20, 21, which is 4 numbers.
2014 · #11 How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used...
How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used more than once in the numbers.
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- We want two-digit numbers made only from 1, 2, 3 that are bigger than 10 and smaller than 32.
- Starting with 1: 11, 12, 13. Starting with 2: 21, 22, 23. Starting with 3 (but under 32): just 31.
- Count the list: 11, 12, 13, 21, 22, 23, 31.
- That makes 7 numbers.
Make pairs with a tree
Some days you mix two things. A shirt AND a pants. You want every outfit you could wear.
You have 2 shirts (red, blue) and 3 pants (white, green, tan). Here is the calm move: take ONE shirt. Match it with each pants. Then take the next shirt and do the same. A little tree shows it.
Count the little lines. Three from red, three from blue. Six lines, six outfits.
Now LOOK at the picture. 2 shirts, and each one made 3 outfits. That is 2 groups of 3. 2 × 3 = 6. The tree showed you why the multiply works.
You drew the tree first. You SAW six lines. Only then did you notice 2 × 3. That is the right order — see it, then shortcut it.
Peek (for older kids)
Take one thing. Match it with all the others. Then take the next thing. Each first thing grows its own little branch.
Do not jump around. Finish the red shirt fully before you start the blue shirt. Then no outfit slips away.
3 girls and 2 boys were dancing. They danced in pairs so that each girl danced with each boy for exactly 1 minute. At any time, there was only one pair on the dance floor. For how many minutes did they dance?
3 girls and 2 boys dance. Each girl dances once with each boy. One pair at a time, 1 minute each. How many minutes do they dance?
Only one pair dances at a time. So the minutes are the same as the number of pairs. This is making pairs!
Take the first girl. She dances with boy 1, then boy 2. That is 2 pairs. Each girl makes 2 pairs. There are 3 girls. So 2 and 2 and 2 is 6 pairs.
Each pair is 1 minute, so they dance 6 minutes.
3 girls, 2 boys each: that is 3 × 2 = 6 pairs, the same as 3 hats and 2 scarves. One pair per minute, so 6 minutes.
To pair two groups: each thing in the first group matches every thing in the second. Draw the tree, count the lines — or multiply the two counts.
2015 · #23 Thomas drew a pig and a shark. He cuts each animal into three pieces. Then he takes one of the two heads, one of the two middle sections...
Thomas drew a pig and a shark. He cuts each animal into three pieces. Then he takes one of the two heads, one of the two middle sections and one of the two tails and lays them together to make another animal. How many different animals can he make in this way?
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- There are 2 heads to choose from, 2 middle sections, and 2 tails.
- Each new animal is one choice for each part, so 2 × 2 × 2 = 8 animals can be built.
- The answer is 8.
2020 · #22 Joana has several sheets of paper, each with a drawing of a parrot. She wants to paint only the head, tail and wing of the parrot, using...
Joana has several sheets of paper, each with a drawing of a parrot. She wants to paint only the head, tail and wing of the parrot, using red, blue or green. The head and tail may be the same colour, but the wing must not be the same colour as the head or the tail. How many sheets can she paint so that no two parrots are painted the same way?
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- If head and tail share a colour (3 ways), the wing has 2 allowed colours: 3×2 = 6.
- If head and tail differ (3×2 = 6 ways), the wing must avoid both, leaving 1 choice: 6.
- Altogether 6 + 6 = 12 different parrots.
Quick check: list and pair
Two warm-ups. Go in order, or draw a little tree.
2014 · #11 How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used...
How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used more than once in the numbers.
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- We want two-digit numbers made only from 1, 2, 3 that are bigger than 10 and smaller than 32.
- Starting with 1: 11, 12, 13. Starting with 2: 21, 22, 23. Starting with 3 (but under 32): just 31.
- Count the list: 11, 12, 13, 21, 22, 23, 31.
- That makes 7 numbers.
2015 · #23 Thomas drew a pig and a shark. He cuts each animal into three pieces. Then he takes one of the two heads, one of the two middle sections...
Thomas drew a pig and a shark. He cuts each animal into three pieces. Then he takes one of the two heads, one of the two middle sections and one of the two tails and lays them together to make another animal. How many different animals can he make in this way?
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- There are 2 heads to choose from, 2 middle sections, and 2 tails.
- Each new animal is one choice for each part, so 2 × 2 × 2 = 8 animals can be built.
- The answer is 8.
Line them up
Now put things in a row, like kids in a line for snack. Here order matters. Who is first is not the same as who is last.
Start as small as it goes. 2 toys: a cat and a dog. Cat then dog. Or dog then cat. That is 2 ways.
Now add one more. 3 toys: cat, dog, fish. Same calm question: who is first? Pick the first one, then line up the leftover two.
Cat first gives 2. Dog first gives 2. Fish first gives 2. 2 and 2 and 2 is 6.
2 toys give 2 ways. 3 toys give 6 ways. See how lining up grows fast? Each new toy could go first, and that doubles your branches.
Pick who goes first. Then line up the rest. Do this for each one who could be first, then add up the branches.
In a line, order matters. cat-dog is NOT the same as dog-cat. Both count! (That is different from picking 2 scoops, where backwards was the same.)
Three boys enter a room one after the other. Hermann is not the first. Felix is not the second. Clemens is not the third. How many different orders are there for the boys to enter the room?
Three boys enter a room one at a time: Hermann, Felix, Clemens. The rules: Hermann is not first, Felix is not second, Clemens is not third. How many orders work?
Three boys line up in 6 ways, the same as our 3 toys. So write all 6 in order, then cross out the bad ones.
- Hermann-Felix-Clemens — no, Hermann is first.
- Hermann-Clemens-Felix — no, Hermann is first.
- Felix-Hermann-Clemens — no, Clemens is third.
- Felix-Clemens-Hermann — yes! ✓
- Clemens-Hermann-Felix — yes! ✓
- Clemens-Felix-Hermann — no, Felix is second.
Only 2 orders are left.
List all 6 orders neatly. Then read each rule and cross out any line that breaks one. Count what is left standing.
To line up: pick who is first, line up the rest, add. If there are rules, list every line and cross out the bad ones.
2015 · #13 Joseph has got a toy car, a teddy bear, a ball and a ship. He wants to put them in a new order on the shelf. The ship must be next to...
Joseph has got a toy car, a teddy bear, a ball and a ship. He wants to put them in a new order on the shelf. The ship must be next to the car, and the teddy bear should also be next to the car. In how many different orders can he put the toys on the shelf?
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- The ship and the teddy both must be next to the car, so the car is in the middle of a block: ship–car–teddy.
- That block can be ordered 2 ways (ship–car–teddy or teddy–car–ship).
- The ball goes at the left end or the right end of the block: 2 choices.
- Total arrangements: 2 × 2 = 4.
2012 · #20 Anna, Laura, Lisa and Katharina wanted to take a photo together. Anna and Katharina are best friends and wanted to stand next to each...
Anna, Laura, Lisa and Katharina wanted to take a photo together. Anna and Katharina are best friends and wanted to stand next to each other. Lisa also wanted to stand next to Anna. In how many different ways can the photo be taken, if all their wishes are met?
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- Anna is next to Katharina and next to Lisa, so the order around Anna is Katharina-Anna-Lisa (or its reverse): 2 ways.
- This block of three can have Laura at the left end or the right end: 2 ways.
- That gives 2 x 2 = 4 line-ups.
- There are 4 ways.
Toys that must stick together
Sometimes two friends MUST stand next to each other in the line. Like best buddies who hold hands. That changes how you count.
Here is the calm trick: if two toys must touch, tape them into one block. Then you have fewer things to line up.
Say a bear and a ball must be side by side, with a car somewhere too. Tape bear and ball together. Now you only line up 2 things: the [bear-ball] block and the car.
The block can go before or after the car: 2 ways. And inside, bear-ball can flip to ball-bear: 2 ways. 2 and 2 makes 4.
So tape the must-touch friends together first. Line up the blocks. Then remember the friends inside can swap places too.
If two things must touch, tape them into one block. Line up the blocks, then let each block flip inside.
Do not forget the flip! [bear-ball] and [ball-ball] look the same on the shelf, but bear-ball and ball-bear are two different orders. Count both.
Joseph has got a toy car, a teddy bear, a ball and a ship. He wants to put them in a new order on the shelf. The ship must be next to the car, and the teddy bear should also be next to the car. In how many different orders can he put the toys on the shelf?
Joseph lines up a car, a teddy, a ball, and a ship. The ship must be next to the car. The teddy must also be next to the car. How many orders?
The car must touch the ship AND the teddy. The only way is for the car to sit in the middle of them: ship-car-teddy. Tape those three into one block.
That block can flip: ship-car-teddy or teddy-car-ship. That is 2 ways.
Now the ball is loose. It goes at the left end or the right end of the block: 2 ways.
So 2 and 2 makes 4 orders.
The car has two friends who both must touch it, so the car is squeezed in the middle. Tape ship-car-teddy as a block (it can flip), then drop the ball on one end. 2 × 2 = 4.
Must-touch friends become one block. Count the block orders, then count the flips inside, then multiply.
2012 · #20 Anna, Laura, Lisa and Katharina wanted to take a photo together. Anna and Katharina are best friends and wanted to stand next to each...
Anna, Laura, Lisa and Katharina wanted to take a photo together. Anna and Katharina are best friends and wanted to stand next to each other. Lisa also wanted to stand next to Anna. In how many different ways can the photo be taken, if all their wishes are met?
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- Anna is next to Katharina and next to Lisa, so the order around Anna is Katharina-Anna-Lisa (or its reverse): 2 ways.
- This block of three can have Laura at the left end or the right end: 2 ways.
- That gives 2 x 2 = 4 line-ups.
- There are 4 ways.
2009 · #21 In a vase there is one red, one blue, one yellow and one white flower. Maja the bee visits each flower exactly once. She begins with the...
In a vase there is one red, one blue, one yellow and one white flower. Maja the bee visits each flower exactly once. She begins with the red flower and she never flies directly from the yellow to the white flower. In how many different ways can she visit each flower?
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- Starting at red, the other three flowers (blue, yellow, white) can be ordered in 6 ways.
- Remove every order in which yellow is immediately followed by white.
- Two of the six orders are forbidden, leaving 4 allowed routes.
- So she can do it in 4 ways.
Quick check: line them up
Pick who goes first. Tape must-touch friends into a block.
2015 · #13 Joseph has got a toy car, a teddy bear, a ball and a ship. He wants to put them in a new order on the shelf. The ship must be next to...
Joseph has got a toy car, a teddy bear, a ball and a ship. He wants to put them in a new order on the shelf. The ship must be next to the car, and the teddy bear should also be next to the car. In how many different orders can he put the toys on the shelf?
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- The ship and the teddy both must be next to the car, so the car is in the middle of a block: ship–car–teddy.
- That block can be ordered 2 ways (ship–car–teddy or teddy–car–ship).
- The ball goes at the left end or the right end of the block: 2 choices.
- Total arrangements: 2 × 2 = 4.
2012 · #20 Anna, Laura, Lisa and Katharina wanted to take a photo together. Anna and Katharina are best friends and wanted to stand next to each...
Anna, Laura, Lisa and Katharina wanted to take a photo together. Anna and Katharina are best friends and wanted to stand next to each other. Lisa also wanted to stand next to Anna. In how many different ways can the photo be taken, if all their wishes are met?
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- Anna is next to Katharina and next to Lisa, so the order around Anna is Katharina-Anna-Lisa (or its reverse): 2 ways.
- This block of three can have Laura at the left end or the right end: 2 ways.
- That gives 2 x 2 = 4 line-ups.
- There are 4 ways.
Count a picture in groups
When you count things in a busy picture, do not poke all over the place. You lose your spot, or count one dot twice. Count in groups instead. Go row by row.
Top row 4. Middle row 4. Bottom row 4. Count by rows and you stay calm.
Touch each row once, in order, and say the running total out loud: four, eight, twelve. Your finger never wanders.
When you make a list, keep it in neat little groups too — all the 1s, then all the 2s. Tidy groups let you SEE if one is missing.
Count in groups, not one-by-one all over. Row by row, or color by color. Then add the groups.
Do not count the same dot twice and do not skip one. Touch each row once, in order, and say the total as you go.
You make two-digit numbers using the digits 2, 0, 1 and 8. Each number must be bigger than 10 and smaller than 25, and made of two different digits. How many different numbers do you get?
Make two-digit numbers from the digits 2, 0, 1, 8. Each must be bigger than 10 and smaller than 25, with two different digits. How many?
Stay tidy. Group by the first digit. Bigger than 10 and under 25 means the number starts with 1 or 2.
Starting with 1: the other digit comes from 2, 0, 8. That gives 12, 10, 18. But 10 is not bigger than 10, so keep 12 and 18.
Starting with 2 and staying under 25: 20 and 21.
All together: 12, 18, 20, 21. That is 4 numbers.
Group by the first digit, the way you count by rows. All the 1-numbers, then all the 2-numbers. Tidy groups, nothing missed.
Count pictures by rows or groups, then add. Keep any list in tidy groups so nothing is missed.
2025 · #3 How many pencils are in this picture?
How many pencils are in this picture?
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- The pencils overlap, so count the ends instead of the middles.
- Match each sharpened dark tip to one pink eraser end.
- There are 8 such pairs, so there are 8 pencils.
2014 · #4 A big square is made from 25 small squares put together. A few of the small squares have been lost. How many have been lost?
A big square is made from 25 small squares put together. A few of the small squares have been lost. How many have been lost?
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- A whole big square is made of 5 × 5 = 25 small squares.
- Carefully count the small squares that are still in the picture: there are 15 of them.
- The lost ones are the ones missing from 25, so 25 − 15 = 10 squares were lost.
2014 · #11 How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used...
How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used more than once in the numbers.
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- We want two-digit numbers made only from 1, 2, 3 that are bigger than 10 and smaller than 32.
- Starting with 1: 11, 12, 13. Starting with 2: 21, 22, 23. Starting with 3 (but under 32): just 31.
- Count the list: 11, 12, 13, 21, 22, 23, 31.
- That makes 7 numbers.
Count what you leave out
Here is a clever turn-around. Sometimes the easy way to count what you want is to count what you do not want.
You have 4 fruits: apple, banana, cherry, grape. You pack 3 in your lunchbox. How many ways?
Listing every group of 3 is fiddly. So flip it. Packing 3 is the same as leaving 1 at home. And there are only 4 fruits to leave!
Leave the apple. Or banana. Or cherry. Or grape. 4 fruits, so 4 ways.
This same flip works when a problem says not. Count all the ways. Then count the not-allowed ways. Take them away.
...you would keep almost everything (count the few you drop instead), or a rule says one thing is not allowed (count all, then subtract the bad ones).
When you keep almost all of them, count the few you leave out. And when a rule says not, count everything, then take the not-allowed ones away.
The leave-out flip only works when the order does not matter (a lunchbox, not a line). In a line, apple-banana-cherry is different from banana-apple-cherry.
A small zoo has a giraffe, an elephant, a lion and a turtle. Susi wants to visit exactly two of the animals today but does not want to start with the lion. How many different possibilities does she have, to visit the two animals one after the other?
The zoo has a giraffe, elephant, lion, and turtle. Susi visits 2 animals, one after the other, and does not start with the lion. How many ways?
The word not is your hint to flip. First count ALL the ways to visit 2 in order. Pick the first animal: 4 choices. Pick the second (a different one): 3 left. That is 4 × 3 = 12 ways.
Now count the bad ones: the visits that DO start with the lion. Lion first, then one of the other 3: that is 3 bad ways.
Take them away: 12 − 3 = 9 ways.
The word not means flip. Count all the ways (4 × 3 = 12), count the ones that break the rule (lion first = 3), then 12 − 3 = 9.
If picking almost all, count the few you leave out. If a rule says not, count all the ways and subtract the bad ones.
2023 · #7 Anna has four discs of different sizes. She wants to build a tower using 3 discs. A smaller disc always has to lie on top of a bigger...
Anna has four discs of different sizes. She wants to build a tower using 3 discs. A smaller disc always has to lie on top of a bigger disc. How many ways are there for Anna to build this tower?
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- Once three discs are chosen, they must go largest-on-bottom, so the order is fixed.
- Thus the number of towers equals the number of ways to pick 3 discs from 4.
- That is the same as choosing which disc to omit: 4 ways, so the answer is 4.
2024 · #14 Lucas has these five puzzle pieces (shown on the right): a smiling head, a banana-tail, and three middle pieces. They snap together only...
Lucas has these five puzzle pieces (shown on the right): a smiling head, a banana-tail, and three middle pieces. They snap together only where a bump fits into a notch. He wants to make a caterpillar with a head, a tail, and 1, 2 or 3 pieces in between. How many different caterpillars can Lucas build?
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- A piece can join another only where a bump fits into a notch, and the head and tail each connect on just one side.
- Try 1 middle piece between head and tail: only the pieces whose bumps and notches line up both ways work.
- Now try 2 middle pieces, then 3 middle pieces, keeping every join a bump-into-notch fit.
- Adding up all the chains that connect properly gives 4 (B) different caterpillars.
2009 · #21 In a vase there is one red, one blue, one yellow and one white flower. Maja the bee visits each flower exactly once. She begins with the...
In a vase there is one red, one blue, one yellow and one white flower. Maja the bee visits each flower exactly once. She begins with the red flower and she never flies directly from the yellow to the white flower. In how many different ways can she visit each flower?
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- Starting at red, the other three flowers (blue, yellow, white) can be ordered in 6 ways.
- Remove every order in which yellow is immediately followed by white.
- Two of the six orders are forbidden, leaving 4 allowed routes.
- So she can do it in 4 ways.
Counting test
Six problems, easy to a little harder. Go in order and take your time!
2025 · #3 How many pencils are in this picture?
How many pencils are in this picture?
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- The pencils overlap, so count the ends instead of the middles.
- Match each sharpened dark tip to one pink eraser end.
- There are 8 such pairs, so there are 8 pencils.
2018 · #12 You make two-digit numbers using the digits 2, 0, 1 and 8. Each number must be bigger than 10 and smaller than 25, and made of two...
You make two-digit numbers using the digits 2, 0, 1 and 8. Each number must be bigger than 10 and smaller than 25, and made of two different digits. How many different numbers do you get?
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- The number is bigger than 10 and smaller than 25, so it starts with 1 or 2.
- Starting with 1, the other digit (from 2, 0, 8) gives 12, 10, 18 — but 10 is not bigger than 10, so keep 12 and 18.
- Starting with 2, staying under 25, gives 20 and 21. All together: 12, 18, 20, 21, which is 4 numbers.
2021 · #21 3 girls and 2 boys were dancing. They danced in pairs so that each girl danced with each boy for exactly 1 minute. At any time, there...
3 girls and 2 boys were dancing. They danced in pairs so that each girl danced with each boy for exactly 1 minute. At any time, there was only one pair on the dance floor. For how many minutes did they dance?
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- There are 3 girls and 2 boys, giving 3 × 2 = 6 different pairs.
- Each pair dances for 1 minute, one pair at a time.
- So they dance for 6 minutes, option B.
2023 · #7 Anna has four discs of different sizes. She wants to build a tower using 3 discs. A smaller disc always has to lie on top of a bigger...
Anna has four discs of different sizes. She wants to build a tower using 3 discs. A smaller disc always has to lie on top of a bigger disc. How many ways are there for Anna to build this tower?
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- Once three discs are chosen, they must go largest-on-bottom, so the order is fixed.
- Thus the number of towers equals the number of ways to pick 3 discs from 4.
- That is the same as choosing which disc to omit: 4 ways, so the answer is 4.
2023 · #19 Three boys enter a room one after the other. Hermann is not the first. Felix is not the second. Clemens is not the third. How many...
Three boys enter a room one after the other. Hermann is not the first. Felix is not the second. Clemens is not the third. How many different orders are there for the boys to enter the room?
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- There are 6 possible orders of three boys.
- Keeping only those with Hermann not first, Felix not second, and Clemens not third leaves Felix-Clemens-Hermann and Clemens-Hermann-Felix.
- That is 2 valid orders.
2009 · #21 In a vase there is one red, one blue, one yellow and one white flower. Maja the bee visits each flower exactly once. She begins with the...
In a vase there is one red, one blue, one yellow and one white flower. Maja the bee visits each flower exactly once. She begins with the red flower and she never flies directly from the yellow to the white flower. In how many different ways can she visit each flower?
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- Starting at red, the other three flowers (blue, yellow, white) can be ordered in 6 ways.
- Remove every order in which yellow is immediately followed by white.
- Two of the six orders are forbidden, leaving 4 allowed routes.
- So she can do it in 4 ways.
Remember this
- Go in order. List slow, smallest first. Never write the same way twice.
- Pairs: draw a little tree — each thing in one group matches every thing in the other.
- Line up: pick who is first, then line up the rest, then add.
- Must touch: tape the friends into one block, and let the block flip.
- Pictures: count by rows or groups, then add.
- Flip it: if you keep almost all, count the one you leave out; if a rule says not, count all and subtract the bad.