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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The given expression is a sum of combinations: $S = ^rC_r + ^{r+1}C_r + ^{r+2}C_r + ...... + ^{n-1}C_r + ^nC_r$.We use the Hockey-Stick Identity (

Vedclass · Accepted Answer

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

Vedclass · Answer

(B) The given expression is a sum of combinations: $S = ^rC_r + ^{r+1}C_r + ^{r+2}C_r + ...... + ^{n-1}C_r + ^nC_r$.We use the Hockey-Stick Identity (also known as the Christmas Stocking Identity),which states that $\sum_{i=r}^{n} {^iC_r} = ^{n+1}C_{r+1}$.Step-by-step application:$1$. Since $^rC_r = 1$ and $^{r+1}C_{r+1} = 1$,we can write $^rC_r$ as $^{r+1}C_{r+1}$.$2$. Using the Pascal's Identity: $^nC_r + ^nC_{r-1} = ^{n+1}C_r$.$3$. Applying this iteratively:$(^{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$$= ^{n+1}C_{r+1}$.Thus,the correct option is $B$.

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

Similar Questions

How many $4$-digit numbers are there with distinct digits?

How many numbers between $5000$ and $10,000$ can be formed using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ if each digit appears not more than once in each number?

The number of ways in which $3$ boys and $4$ girls can be seated around a circular table if no two boys sit together is:

On the sides $AB, BC, CA$ of a $\Delta ABC$,$3, 4, 5$ distinct points (excluding vertices $A, B, C$) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are:

In how many ways can $5$ keys be put in a ring?

Vedclass Test Series

Exam Paper Generator

Online Exam Module