Problem 18 · 2017 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
The numbers 1, 2, 3, 4 and 5 are to be written into the five cells of this diagram by the following rules: if one number is below another it must be greater, and if one number is to the right of another it must be greater. How many ways are there to place the numbers?
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Answer: D — 6
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Hint 1 of 2
Numbers must increase going right and increase going down, which forces 1 into the top-left and 5 into the bottom-right.
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Hint 2 of 2
Count the valid fillings of the remaining cells respecting both rules.
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Approach: place the forced smallest and largest, then list the few valid fillings
- Every cell must be bigger than the one to its left and the one above it, so the very top-left cell is the smallest, 1, and the very bottom-right cell is the largest, 5.
- Now fill the remaining cells with 2, 3 and 4, always keeping each one larger than its left and upper neighbours.
- Going through the possibilities carefully, exactly 6 different fillings obey both rules.
- So there are 6 ways (D).
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