Problem 24 · 2020 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
The Kangaroo Hotel has 30 floors, numbered 1 to 30, and each floor has 20 rooms, numbered 1 to 20. The code to enter a room is formed by writing the floor number followed by the room number, in that order. But a code can be confusing: for example, the code 111 could mean floor 11 room 1 or floor 1 room 11. Note that the code 101 is not confusing, since it can only mean floor 10 room 1 (floor 1 room 1 has the code 11, not 101). How many codes are confusing, including the one in the example?
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Answer: E — 18
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Hint 1 of 3
A code is confusing when the same digits can be cut into a floor and a room in two different correct ways.
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Hint 2 of 3
For three digits, you can cut after the first digit or after the second digit, so look for codes where both cuts give a real floor and a real room.
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Hint 3 of 3
Think about which three-digit codes start with a small floor but could also start with a teens floor.
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Approach: find three-digit codes that can be cut two ways, both giving a real floor and room
- A three-digit code can be cut after the 1st digit (1-digit floor, 2-digit room) or after the 2nd digit (2-digit floor, 1-digit room); it is confusing when BOTH cuts give a real floor (1 to 30) and room (1 to 20).
- For both cuts to work, the middle digit must be 1, so the codes look like floor-1-room, and they are exactly 11c and 21c where c is 1 to 9.
- 11c reads as floor 1 room 1c or floor 11 room c, and 21c reads as floor 2 room 1c or floor 21 room c, and c can be 1 to 9.
- That is 9 codes of the form 11c and 9 codes of the form 21c, so 9 + 9 = 18 confusing codes, choice E.
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