In the expansion of ${(1 + x)}^{n} {(1 + y)}^{n} {(1 + z)}^{n}$ , then the sum of coefficients of the terms of degree $m$ is

{({^{n} C}_{m})}^{3}

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3. {({^{n} C}_{m})}^{3}

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^{3 n} C_{m}

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{({^{n} C}_{m})}^{3 m}

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Solution

The correct option is B $^{3 n} C_{m}$
${(1 + x)}^{n} {(1 + y)}^{n} {(1 + z)}^{n}$
$= (n \sum r = 0^{n} C_{r} x^{r}) (n \sum s = 0^{n} C_{s} y^{s}) (n \sum t = 0^{n} C_{t} z^{t})$
$= \sum 0 \leq r, s, t \leq n (^{n} C_{r}) (^{n} C_{s}) (^{n} C_{t}) x^{r} y^{s} z^{t}$
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 $= \sum r, s, t \geq 0 r + s + t = m (^{n} C_{r}) (^{n} C_{s}) (^{n} C_{t})$
$=$ the number of ways of choosing a total number of $m$ balls out of $n$ black, $n$ white and $n$ green balls
$=^{3 n} C_{m}$

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