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  4. nC0 + n+1C1 + n+2C2 + .... + n+rCr is equal to

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Q. $^nC_0\,+\,{}^{n+1}C_1\,+\,{}^{n+2}C_2\,+\,....\,+\,{}^{n+r}C_r$ is equal to

Permutations and Combinations

Solution:

Given expression
$\left(^{n+1}C_{0} +\,{}^{n+1}C_{1}\right)+\,{}^{n+2}C_{2}+ \,{}^{n+3}C_{3} +.....+\,{}^{n+r}C_{r} $
$\left[\because\,{}^{n}C_{0}=^{n+1}C_{0}\right]$
$=\left(^{n+2}C_{1}+\,{}^{n+2}C_{2}\right)+ \,{}^{n+3}C_{3} + ....+\,{}^{n+r}C_{r}$
$=\left(^{n+3}C_{2}+\,{}^{n+3}C_{3}\right)+....+ \,{}^{n+r}C_{r}$
$=\left(^{n+4}C_{3}+\,{}^{n+4}C_{4}\right) + ....+\,{}^{n+r}C_{r}$
$=\,{}^{n+r}C_{r-1}+\,{}^{n+r}C_{r}=\,{}^{n+r+1}C_{r}$