Solution: 2018 Winter Final - 6

Author: Michiel Smid

Question

Nick (your friendly TA) 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)

${31 \choose 7}$

(c)

${31 \choose 6}$

(d)

${32 \choose 6}$

Solution

We can rephrase this into something more familiar. 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 $

This is one of the divider questions

Assume we have 25 bananas and 6 dividers. We place the dividers between any set of bananas to indicate how many are eaten between dividers

$x_1$ is represented as the number of bananas eaten to the left of the first divider
$x_2$ is represented as the number of bananas eaten between the first and second divider
$x_3$ is represented as the number of bananas eaten between the second and third divider
$x_4$ is represented as the number of bananas eaten between the third and fourth divider
$x_5$ is represented as the number of bananas eaten between the fourth and fifth divider
$x_6$ is represented as the number of bananas eaten between the fifth and sixth divider
$x_7$ is represented as the number of bananas eaten to the right of the sixth divider

Since we place dividers down, we need to add 6 dividers to the 25 bananas

We choose 6 positions out of the 31 for the dividers: $ \binom{31}{6} $