Solution: 2018 Winter Midterm - 9

Author: Michiel Smid

Question

Consider the equation $$ x_1 + x_2 + x_3 + x_4 = 33, $$ where $x_1 \geq 0$, $x_2 \geq 0$, $x_3 \geq 0$, $x_4 \geq 0$ are integers. How many solutions does this equation have?

(a)

${36 \choose 4}$

(b)

${37 \choose 3}$

(c)

${36 \choose 3}$

(d)

${37 \choose 4}$

Solution

We can use the dividers method to solve this problem.

We have 33 blocks (representing the sum of the 4 variables) and 3 dividers (representing the 3 partitions between the 4 variables).

We can place the 3 dividers into any of the 36 positions.

Everything before the first divider is $ x_1 $, everything between the first and second dividers is $ x_2 $, and so on.

Thus, there are $ \binom{36}{3} $ solutions to the equation.