Problem 27 · 2024 Math Kangaroo
Stretch
Counting & Probability
sum-constraintcasework
Ann rolled an ordinary die 24 times. Every number from 1 to 6 came up at least once, and the number 1 came up more often than any other number. Ann then added all the numbers she rolled. What is the largest total she could have obtained?
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Answer: D — 90
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Hint 1 of 2
Every face appears at least once, and 1 must appear strictly more often than each other face.
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Hint 2 of 2
Allow 1 to appear enough times that the other faces may each appear up to one less, then load 5's and 6's.
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Approach: minimise the count of 1 then maximise the high faces
- Let 1 appear c times; every other face appears at most c-1 times, and all six faces appear at least once.
- To maximise the sum we want many 5's and 6's, so allow c = 7: then faces can appear up to 6 times each, e.g. six 6's, six 5's, three 4's, one 3, one 2, and seven 1's (total 24 rolls).
- That total is 7*1 + 1*2 + 1*3 + 3*4 + 6*5 + 6*6 = 90.
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