Sum of the coefficients of the terms of degree m in the expansion of (1+x)n(1+y)n(1+z)n is

A
${(^{n} C_{m})}^{3}$
B
$3 (^{n} C_{m})$
C
$^{n} C_{3 m}$
D
$^{3 n} C_{m}$

Solution:

We have
$\begin{array}{l} (1 + x)^{n} (1 + y)^{n} (1 + z)^{n} \\ = (\sum_{r = 0}^{n}^{n} C_{r} x^{r}) (\sum_{s = 0}^{n}^{n} C_{s} y^{s}) (\sum_{t = 0}^{n}^{n} C_{r} z^{t}) \\ = \sum_{0 \leq r, s, l \leq n} (^{n} C_{r}) (^{n} C_{s}) (^{n} C_{l}) x^{r} y^{s} z^{t} \end{array}$
For sum of the coefficients of degree $m$ , we must have
$r + s + t = m$ where $r, s, t$ are integers with $r, s, t \geq 0 .$
Sum of such coefficients
$= \sum_{\binom{r, s, l \geq 0}{r + s + r = m}} (^{n} C_{r}) (^{n} C_{s}) (^{n} C_{t})$
$=$ the number of ways of choosing a total number of m balls out of $n$ distinct black, n distinct white and $n$ distinct green balls
$=^{3 n} C_{m}$

Sum of the coefficients of the terms of degree m in the expansion of $(1 + x)^{n} (1 + y)^{n} (1 + z)^{n}$ is

Solution:

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