Problem:
When fair standard -sided dice are thrown, the probability that the sum of the numbers on the top faces is can be written as
where is a positive integer. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The only ways to achieve a sum of by adding unordered integers between and inclusive are (i) six s and one ; (ii) five s, one , and one ; or (iii) four s and three s. The number of ways to order the outcomes among the dice are in case (i), in case (ii), and in case (iii). There are possible outcomes. Therefore .
The number of ways to achieve a sum of by adding ordered integers between and , inclusive, is the same as the number of ways to insert bars in the spaces between stars in a row of stars (with no more than one bar per space). For example, the sum corresponds to . The number of ways of inserting bars in the spaces in a row of stars is . (This approach is referred to as "stars and bars".)
The problems on this page are the property of the MAA's American Mathematics Competitions