2018 AMC 10A Problem 11

Question

When

7

fair standard

6

-sided dice are thrown, the probability that the sum of the numbers on the top faces is

10

can be written as

\dfrac{n}{6^{7}},

where

n

is a positive integer. What is

n?

42

49

56

63

84

List the unordered ways seven positive die rolls can sum to

10

Count the orderings for each listed pattern

We can use stars and bars to find

n.

It is the same as finding the number of ways to put

10

balls into

7

boxes, where each box has at least one ball.

The formula for such a scenario is

\binom{n - 1}{k - 1},

where

n

is the number of balls and

k

is the number of boxes.

No die can exceed

4

in a sum of

10

from seven positive rolls, so the upper bound of

6

creates no additional restriction. The desired answer is therefore

\binom{9}{6} = \binom{9}{3} = 84.

Thus, E is the correct answer.

Problem 11 in other years