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

$^{3 n} C_{r}$

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${3.}^{n} C_{r}$

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$^{3 n} C_{3 r}$

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$^{3 n} C_{3 n - r}$

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Solution

The correct options are
A
$^{3 n} C_{r}$

D
$^{3 n} C_{3 n - r}$

$(1 + x)^{n} (1 + y)^{n} (1 + z)^{n} (\sum_{m = 0}^{n}^{n} C_{m} x^{m}) (\sum_{s = 0}^{n}^{n} C_{s} y^{s}) (\sum_{t = 0}^{n}^{n} C_{t} z^{t}) = \sum_{0 \leq r, s, t \leq n}^{n} C_{m}^{n} C_{s}^{n} C_{t} X^{m} Y^{s} Z^{t}$

For sum of the coefficients of degree ‘r’, we have m + s + t = r; where, m, s, t are integers greater than or equal to zero. Sum of such coefficients

$= {\sum \begin{matrix} m, s, t \geq 0 m + s + t = r \end{matrix}}^{n} C_{m}^{n} C_{s}^{n} C_{t}$ (this is similar to the number of ways of choosing a total of r balls out of n black, n white, n green balls.) $=^{3 n} C_{r}$

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Similar questions

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

(1 + x)^{n} . (1 + y)^{n} . (1 + z)^{n}

r

m

{(1 + x)}^{n} {(1 + y)}^{n} {(1 + z)}^{n}

(r + 1)^{t h}

(1 + x)^{n + 1}

r^{t h}

{r + 1}^{t h}

(1 + x)^{n}

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