2016 AIME I Problems/Problem 8
Problem 8
For a permutation of the digits , let denote the sum of the three -digit numbers , , and . Let be the minimum value of subject to the condition that the units digit of is . Let denote the number of permutations with . Find .
Solution
To minimize , the numbers , , and (which sum to ) must be in the hundreds places. For the units digit of to be , the numbers in the ones places must have a sum of either or . However, since the tens digit contributes more to the final sum than the ones digit, and we are looking for the minimum value of , we take the sum's units digit to be . We know that the sum of the numbers in the tens digits is . Therefore, .
To find , realize that there are ways of ordering the numbers in each of the places. Additionally, there are three possibilities for the numbers in the ones place: , , and . Therefore there are ways in total. .
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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