First slide

Combinations

Question

The number of integers $x, y, z, w$ such that $x + y + z + w = 20$ and $x, y, z, w \geq - 1,$ is

Moderate

A

$^{24} C_{3}$
B

$^{25} C_{3}$
C

$^{26} C_{3}$
D

$^{27} C_{3}$

Solution

Put $x = a - 1, y = b - 1, z = c - 1, w = d - 1$
then $a, b, c, d \geq 0 and (a - 1) + (b - 1) + (c - 1) + (d - 1) = 20$
$\Rightarrow a + b + c + d = 24$
The number of non-negative integral solutions of this equation
is
$^{24 + 4 - 1} C_{4 - 1} =^{27} C_{3}$

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