^nC_r + ^{n-1}C_r + ...... + ^rC_r = | vedclass

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(B) The given expression is $^rC_r + ^{r+1}C_r + ^{r+2}C_r + ...... + ^{n-1}C_r + ^nC_r$.Using the identity $^nC_r + ^nC_{r-1} = ^{n+1}C_r$,we know th

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$^{n+1}C_{r+1}$

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(B) The given expression is $^rC_r + ^{r+1}C_r + ^{r+2}C_r + ...... + ^{n-1}C_r + ^nC_r$.Using the identity $^nC_r + ^nC_{r-1} = ^{n+1}C_r$,we know that $^rC_r = ^{r+1}C_{r+1}$.Substituting this,we get:$^{r+1}C_{r+1} + ^{r+1}C_r + ^{r+2}C_r + ...... + ^nC_r$$= ^{r+2}C_{r+1} + ^{r+2}C_r + ...... + ^nC_r$$= ^{r+3}C_{r+1} + ...... + ^nC_r$Continuing this process,the sum simplifies to:$^{n}C_{r+1} + ^nC_r = ^{n+1}C_{r+1}$.

$^nC_r + ^{n-1}C_r + ...... + ^rC_r =$

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