The value of sumlimits_{r = 0}^{15} {left( {{ | vedclass

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

Question

(A) We use the identity ${}^{n}{C_r} = {}^{n}{C_{n-r}}$.The expression is $\sum\limits_{r = 0}^{15} {\left( {}^{15}{C_r} \cdot {}^{40}{C_{15}} \cdot {

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) We use the identity ${}^{n}{C_r} = {}^{n}{C_{n-r}}$.The expression is $\sum\limits_{r = 0}^{15} {\left( {}^{15}{C_r} \cdot {}^{40}{C_{15}} \cdot {}^{20}{C_r} - {}^{35}{C_{15}} \cdot {}^{15}{C_r} \cdot {}^{25}{C_r} \right)}$.Rewrite the terms using the identity ${}^{20}{C_r} = {}^{20}{C_{20-r}}$ and ${}^{25}{C_r} = {}^{25}{C_{25-r}}$:$= {}^{40}{C_{15}} \sum\limits_{r = 0}^{15} {({}^{15}{C_r} \cdot {}^{20}{C_{20-r}})} - {}^{35}{C_{15}} \sum\limits_{r = 0}^{15} {({}^{15}{C_r} \cdot {}^{25}{C_{25-r}})}$.Using Vandermonde's Identity,$\sum\limits_{k=0}^{r} {}^{m}{C_k} \cdot {}^{n}{C_{r-k}} = {}^{m+n}{C_r}$:$= {}^{40}{C_{15}} \cdot {}^{35}{C_{20}} - {}^{35}{C_{15}} \cdot {}^{40}{C_{25}}$.Since ${}^{35}{C_{20}} = {}^{35}{C_{15}}$ and ${}^{40}{C_{25}} = {}^{40}{C_{15}}$:$= {}^{40}{C_{15}} \cdot {}^{35}{C_{15}} - {}^{35}{C_{15}} \cdot {}^{40}{C_{15}} = 0$.

The value of $\sum\limits_{r = 0}^{15} {\left( {{}^{15}{C_r} \cdot {}^{40}{C_{15}} \cdot {}^{20}{C_r} - {}^{35}{C_{15}} \cdot {}^{15}{C_r} \cdot {}^{25}{C_r}} \right)}$ is:

Similar Questions

The number of all five-letter words (with or without meaning) having at least one repeated letter that can be formed by using the letters of the word $INCONVENIENCE$ is:

Let $n = 1! + 4! + 7! + \ldots + 400!$. Then the ten's digit of $n$ is

Numbers between $1$ and $10,000$ are formed using the digits $2$ and $3$ exactly once and the digit $4$ twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of $4324$ and $324$ respectively,then $x-y=$

The number of seven-digit integers,with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only,is

If $\sum\limits_{r = 0}^{25} {\left( {^{50}C_r \cdot ^{50 - r}C_{25 - r}} \right) = K\left( {^{50}C_{25}} \right)}$,then $K$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module