First slide

Combinations

Question

The number of integral solutions of $x + y + z = 0$ with $x \geq - 5, y \geq - 5, z \geq - 5$ is

Moderate

A

134
B

136
C

138
D

140

Solution

Let $x_{1} = x + 5, y_{1} = y + 5, z_{1} = z + 5$ and find the number of non-negative integral solutions of $x_{1} + y_{1} + z_{1} = 15$ .
$15 + 3 - 1_{C_{3}} = 17_{C_{3}} = 136$

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