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Q.
The A.M . of the coefficient in the expansion of $(1+x)^{30}$ is
Statistics
Solution:
Consider $(1+x)^{30}=C_{0}+C_{1}x+C_{2}x_{2}+\dots+C_{30}x^{30}$
Putting $x = 1$, we get
$C_{0}+C_{1}+C_{2}+\dots + C_{30}=2^{30}$
Now , number of terms which are the coefficients of various terms in the expansion of $(1+x)^{30}$ are $31$
$\therefore $ Mean $=\frac{C_{0}+C_{1}+C_{2}+\ldots+C_{30}}{31}=\frac{2^{30}}{31}$
$=\frac{2^{n}}{\left(n+1\right)}$, if $n=30$ Alternative Solution :
The coefficients in the expansion of $(1+x)^{30}$ are $^{30}C_{0}, ^{30}C_{1}, ^{30}C_{2}, \ldots, ^{30}C_{30} $
$(\because$ Number of coefficients are 31)
$\therefore \frac{^{30}C_{0}+^{30}C_{1}+^{30}C_{2}+\ldots+^{30}C_{30}}{31}=\frac{2^{30}}{31}$
$=\frac{2^{n}}{n+1}$ $(if \,n=30)$