Anne ruled the normal die 24 times. All numbers from 1 to 6 came up at least once. The number 1 came up more times than any other number. What is the largest possible sum Anne could have obtained by adding up all the numbers? We know that Anne ruled each number at least once, so we know that we at least have a sum of 21 using 1 through 6 each one time. Let's think about the remaining 18 rules. We know that there should be more 1s than any other number, and then it's clear that the number of 6s ruled should be one less than the number of 1s in order to maximize the sum. If we have 9 1s, then we have 8 6s and 1 5. This gives a sum of 62. If we have 8 1s, then we can have 7 6s and 3 5s. This gives a sum of 65. If we have 7 1s, then we can have 6 6s and 5 5s. This gives a sum of 68. With 6 1s, we can have 5 6s, 5 5s, and 2 4s. Then this gives a sum of 69. This is the highest possible sum that we can find using these 18 rules, and we can check that going with 5 1s and 4 1s will not yield a better answer, and therefore we can take the original 21 and add 69, giving 90, giving our answer as D.

dice

maximum sum

probability

number distribution

mathematics