The number of ways of choosing 10 objects out | vedclass

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

Question

(A) Let the $10$ identical objects be $I$ and the $21$ distinct objects be $D_1, D_2, ..., D_{21}$.We need to choose $10$ objects in total.Let $k$ be

Vedclass · Accepted Answer

$2^{20}$

Vedclass · Answer

(A) Let the $10$ identical objects be $I$ and the $21$ distinct objects be $D_1, D_2, ..., D_{21}$.We need to choose $10$ objects in total.Let $k$ be the number of distinct objects chosen,where $0 \le k \le 10$.The number of ways to choose $k$ distinct objects from $21$ is $^{21}C_k$.Once $k$ distinct objects are chosen,the remaining $(10-k)$ objects must be chosen from the $10$ identical objects. Since the objects are identical,there is only $1$ way to choose them.Thus,the total number of ways is the sum of $^{21}C_k$ for $k=0$ to $10$:Total ways $= \sum_{k=0}^{10} {^{21}C_k}$.We know that $\sum_{k=0}^{21} {^{21}C_k} = 2^{21}$.Also,by symmetry,$\sum_{k=0}^{10} {^{21}C_k} = \sum_{k=11}^{21} {^{21}C_k}$.Let $S = \sum_{k=0}^{10} {^{21}C_k}$. Then $2S = \sum_{k=0}^{10} {^{21}C_k} + \sum_{k=11}^{21} {^{21}C_k} = \sum_{k=0}^{21} {^{21}C_k} = 2^{21}$.Therefore,$S = \frac{2^{21}}{2} = 2^{20}$.

The number of ways of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is

Similar Questions

In how many ways can $5$ speakers $S_1, S_2, S_3, S_4$,and $S_5$ give speeches one after the other if $S_3$ wants to speak after $S_1$ and $S_2$?

The number of rectangles that can be obtained by joining four of $12$ vertices of a $12$-sided regular polygon is:

How many numbers consisting of $5$ digits can be formed in which the digits $3, 4$ and $7$ are used only once and the digit $5$ is used twice?

In how many ways can $15$ members of a council sit along a circular table,if the Secretary must sit on one side of the Chairman and the Deputy Secretary on the other side?

How many numbers greater than a million can be formed with the digits $2, 3, 0, 3, 4, 2, 3$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module