Solution: 2014 Fall Final - 9
Author: Michiel SmidQuestion
How many solutions are there to the equation $x_1 + x_2 + x_3 = 13$, where $x_1 \geq 0$, $x_2 \geq 0$,
$x_3 \geq 0$ are integers?
(a)
${14 \choose 2}$
(b)
${13 \choose 2}$
(c)
${15 \choose 2}$
(d)
${16 \choose 2}$
Solution
We can solve this using stars and bars approach
There are 13 stars and 2 extra positions to place the bars/dividers between rows of stars
- The number of stars to the left of the first bar is $x_1$
- The number of stars between the first and second bars is $x_2$
- The number of stars to the right of the second bar is $x_3$
As a result, we pick 2 positions out of 13 and 2 positions to place the bars: $ \binom{15}{2} $