Solution: 2015 Winter Midterm - 9
Author: Michiel SmidQuestion
How many solutions are there to the equation $x_1 + x_2 + x_3 = 99$, where $x_1 \geq 0$, $x_2 \geq 0$,
and $x_3 \geq 0$ are integers?
(a)
${102 \choose 3}$
(b)
${102 \choose 2}$
(c)
${101 \choose 2}$
(d)
${101 \choose 3}$
Solution
We can use the dividers method.
We have 99 buckets and 2 extra spots for the dividers to go.
Thus, there are $\binom{99+2}{2}$ solutions.