Solution: 2016 Fall Final - 11

Author: Michiel Smid

Question

How many solutions are there to the equation $$ x_1 + x_2 + x_3 + x_4 + x_5 = 28, $$ where $x_1 \geq 0$, $x_2 \geq 0$, $x_3 \geq 0$, $x_4 \geq 0$, and $x_5 \geq 0$ are integers?

(a)

${32 \choose 4}$

(b)

${33 \choose 5}$

(c)

${32 \choose 5}$

(d)

${33 \choose 4}$

Solution

We can use the stars and bars method to solve this problem.

We have 4 dividers and 28 stars.

$x_1$ is the number of stars to the left of the first divider
$x_2$ is the number of stars between the first and second dividers
$x_3$ is the number of stars between the second and third dividers
$x_4$ is the number of stars between the third and fourth dividers
$x_5$ is the number of stars to the right of the fourth divider

The number of solutions to the equation is $ \binom{28+4}{4} $