The sum of the coefficients of the terms of d | vedclass

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Question

(D) The expansion is given by $(1+x)^n(1+y)^n(1+z)^n = \sum_{r=0}^n {}^nC_r x^r \sum_{s=0}^n {}^nC_s y^s \sum_{t=0}^n {}^nC_t z^t = \sum_{r,s,t} ({}^n

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$({}^{3n}C_m)$

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(D) The expansion is given by $(1+x)^n(1+y)^n(1+z)^n = \sum_{r=0}^n {}^nC_r x^r \sum_{s=0}^n {}^nC_s y^s \sum_{t=0}^n {}^nC_t z^t = \sum_{r,s,t} ({}^nC_r {}^nC_s {}^nC_t) x^r y^s z^t$.The terms of degree $m$ are those for which $r + s + t = m$,where $0 \le r, s, t \le n$.The sum of the coefficients of these terms is the sum of the products ${}^nC_r {}^nC_s {}^nC_t$ over all non-negative integers $r, s, t$ such that $r + s + t = m$.This is equivalent to the coefficient of $x^m$ in the expansion of $(1+x)^n(1+x)^n(1+x)^n = (1+x)^{3n}$.The coefficient of $x^m$ in $(1+x)^{3n}$ is given by ${}^{3n}C_m$.

The sum of the coefficients of the terms of degree $m$ in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is:

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