The value of sum_{r=0}^{20} {}^{50-r}C_{6} is | vedclass

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

Question

(B) The given sum is $S = \sum_{r=0}^{20} {}^{50-r}C_{6} = {}^{50}C_{6} + {}^{49}C_{6} + {}^{48}C_{6} + \dots + {}^{30}C_{6}$.Using the Hockey-stick i

Vedclass · Accepted Answer

${}^{51}C_{7} - {}^{30}C_{7}$

Vedclass · Answer

(B) The given sum is $S = \sum_{r=0}^{20} {}^{50-r}C_{6} = {}^{50}C_{6} + {}^{49}C_{6} + {}^{48}C_{6} + \dots + {}^{30}C_{6}$.Using the Hockey-stick identity,which states that $\sum_{i=r}^{n} {}^{i}C_{r} = {}^{n+1}C_{r+1}$,we can rewrite the sum.First,note that ${}^{30}C_{6} = {}^{31}C_{7} - {}^{30}C_{7}$.Alternatively,we can use the property ${}^{n}C_{r} + {}^{n}C_{r+1} = {}^{n+1}C_{r+1}$ repeatedly:$S = {}^{50}C_{6} + {}^{49}C_{6} + \dots + {}^{31}C_{6} + {}^{30}C_{6}$.Adding and subtracting ${}^{30}C_{7}$:$S = ({}^{50}C_{6} + {}^{49}C_{6} + \dots + {}^{30}C_{6} + {}^{30}C_{7}) - {}^{30}C_{7}$.Using the identity ${}^{n}C_{r} + {}^{n}C_{r+1} = {}^{n+1}C_{r+1}$ iteratively:${}^{30}C_{6} + {}^{30}C_{7} = {}^{31}C_{7}$.${}^{31}C_{6} + {}^{31}C_{7} = {}^{32}C_{7}$.Continuing this process up to ${}^{50}C_{6} + {}^{50}C_{7} = {}^{51}C_{7}$.Thus,the sum simplifies to ${}^{51}C_{7} - {}^{30}C_{7}$.

The value of $\sum_{r=0}^{20} {}^{50-r}C_{6}$ is equal to

Similar Questions

In a certain test,there are $n$ questions. In the test,$2^{n-i}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, \dots, n$. If the total number of wrong answers given is $2047$,then $n$ is equal to

How many different numbers of $6$ digits can be formed with the digits $3, 1, 7, 0, 9, 5$?

In how many ways can $10$ true-false questions be replied?

If $a, b$ and $c$ are the greatest values of $^{19}C_{p}, ^{20}C_{q}$ and $^{21}C_{r}$ respectively,then

In how many ways can a football team of $11$ players be selected from $15$ players? In how many cases can a particular player be included?

Vedclass Test Series

Exam Paper Generator

Online Exam Module