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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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