Solution: 2019 Fall Final - 7

Author: Michiel Smid

Question

Nick eats lots of bananas. During a period of 7 days, Nick eats a total of 25 bananas. A banana schedule is a sequence of 7 numbers, whose sum is equal to 25, and whose numbers indicate the number of bananas that Nick eats on each day. Three examples of such schedules are (3,2,7,4,1,3,5), (2,3,7,4,1,3,5), and (3,0,9,4,1,0,8). How many banana schedules are there?

(a)

${32 \choose 7}$

(b)

${31 \choose 6}$

(c)

${31 \choose 7}$

(d)

${32 \choose 6}$

Solution

Given the fact that we all have multiple unknown variables that we need to solve for that will all have to sum to a specific numbers, we can use the formula from “Counting Solutions of Linear Equations” or Chapter 3.9.2 in the textbook to solve this problem: $\binom{n+k-1}{k-1}$

Let $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ be the number of bananas Nick eats on each day: $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 25$

In this case, let $n$ represent the number of bananas Nick eats and $k$ represent the number of days/number of unknown variables. Therefore, $n = 25$ and $k = 7$. Now we can invoke the formula to solve for the number of solutions:

$\binom{25+7-1}{7-1} = \binom{31}{6}$