Question

The A.M . of the coefficient in the expansion of $(1+x{)}^{30}$ is

• A. $\frac{2^n}{n+1}$(if n=30)
• B. $\frac{2^{30}}{30}$
• C. $\frac{2^{29}}{11}$
• D. None of these

Answer 1

Answer: (A)

Consider $(1+x{)}^{30}=C_0+C_{1}x+C_2{x}_2+\cdots +C_{30}{x}^{30}$
Putting $x=1$, we get
$C_0+C_1+C_2+\cdots +C_{30}=2^{30}$
Now , number of terms which are the coefficients of various terms in the expansion of $(1+x{)}^{30}$ are $31$
$\therefore$ Mean $=\frac{C_0+C_1+C_2+\dots +C_{30}}{31}=\frac{2^{30}}{31}$
$=\frac{2^n}{\left(n+1\right)}$, if $n=30$
Alternative Solution :
The coefficients in the expansion of $(1+x{)}^{30}$ are ${}^{30}{C}_0{,}^{30}{C}_1{,}^{30}{C}_2,\dots {,}^{30}{C}_{30}$
$(\because$ Number of coefficients are 31)
$\therefore \frac{{}^{30}{C}_0{+}^{30}{C}_1{+}^{30}{C}_2+\dots {+}^{30}{C}_{30}}{31}=\frac{2^{30}}{31}$
$=\frac{2^n}{n+1}$ $(ifn=30)$