Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The value of <img alt="{ }^n C_0 \times{ }^{2 n} C_n-{ }^n C_1 \times{ }^{2 n-1} C_n+{ }^n C_2 \times{ }^{2 n-2} C_n-\ldots+\\ (-1)^n \times{ }^n C_n \times{ }^n C_n" src="https://entrancecorner.on

The value of { }^n C_0 \times{ }^{2 n} C_n-{ }^n C_1 \times{ }^{2 n-1} C_n+{ }^n C_2 \times{ }^{2 n-2} C_n-\ldots+\\ (-1)^n \times{ }^n C_n \times{ }^n C_n is

Option: 1

0


Option: 2

1


Option: 3

-1


Option: 4

2


Answers (1)

best_answer

We know that
{ }^{2 n} C_n=$ coefficient of $x^n$ in $(1+x)^{2 n}

\begin{aligned} & { }^{2 n-1} C_n=\text { coefficient of } x^n \text { in }(1+x)^{2 n-1} \\ & { }^{2 n-2} C_n=\text { coefficient of } x^n \text { in }(1+x)^{2 n-2} \\ & { }^{2 n-3} C_n=\text { coefficient of } x^n \text { in }(1+x)^{2 n-3} \\ & \vdots \\ & { }^n C_n=\text { coefficient of } x^n \text { in }(1+x)^n \end{aligned}\\\ \text{Thus, we have}\\\\ \begin{array}{r} { }^n C_0 \times{ }^{2 n} C_n-{ }^n C_1 \times{ }^{2 n-1} C_n+{ }^n C_2 \times{ }^{2 n-2} C_n \\ -\ldots+(-1)^n \times{ }^n C_n \times{ }^n C_n \end{array}\\\ \text{= coefficient of }x^n in\\\\ \begin{aligned} & {\left[{ }^n C_0(1+x)^{2 n}-{ }^n C_1(1+x)^{2 n-1}+{ }^n C_2(1+x)^{2 n-2}\right.} \\ & \left.-{ }^n C_3(1+x)^{2 n-3}+\ldots+(-1)^{n n} C_n(1+x)^n\right] \end{aligned}= \text{coefficient of}\ x^n \ in\\\\ \begin{aligned} & \begin{array}{r} (1+x)^n\left[{ }^n C_0(1+x)^n-{ }^n C_1(1+x)^{n-1}+{ }^n C_2(1+x)^{n-2}\right. \\ \left.-{ }^n C_3(1+x)^{n-3}+\ldots+(-1)^{n n} C_n\right] \end{array} \\ = & \text { coefficient of } x^n \text { in }(1+x)^n[(1+x)-1]^n \\ = & \text { coefficient of } x^n \text { in }(1+x)^n x^n=1 \end{aligned}

Posted by

Devendra Khairwa

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026 Answer Key LIVE: NTA to release session 2 answer key tomorrow; result by April 20
anam-khan

NTA JEE Main 2026 session 2 ends with 92.1% attendance; paper 1 answer key on April 11
anam-khan

JEE Main 2026 Session 2 concludes; answer key on April 11, result on 20

Latest Question