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)

${31 \choose 7}$

(b)

${32 \choose 7}$

(c)

${32 \choose 6}$

(d)

${31 \choose 6}$

Solution

Yeah, so this is the dividers method. Let’s define some stuff I guess

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

Now, $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 25$

Assuming you have a sea of bananas, let’s put dividers to split them off into different days

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

Now, we have 6 dividers and 25 bananas, so we have 31 objects in total

Since each object has a position, we can just move the dividers around to change any x.

We choose 6 spots out of the 31 to be dividers: $ \binom{31}{6} $