Question

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. ${\left({{}^{n}C}_m\right)}^3$
• B. $3\left({{}^{n}C}_m\right)$
• C. ${{}^{n}C}_{3m}$
• D. ${{}^{3n}C}_m$

Answer 1

The correct option is D ${{}^{3n}C}_m$

The Coefficient of $x^m$ $=$ Number of ways of choosing $m$ balls out of $n$ black balls, $n$ green balls and $n$ blue ball.
Hence total number of balls $=3n$.
Required is $m$.
Hence required combination is ${}^{3n}{C}_m$.
Hence the coefficient of $x^m$ in $(1+x{)}^n(1+y{)}^n(1+z{)}^n$
$={}^{3n}{C}_m$.