If n is even and the value of ^nC_r is maximu | vedclass

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

Question

(A) The binomial coefficient series is given by $^nC_0, ^nC_1, ^nC_2, \ldots, ^nC_r, \ldots, ^nC_{n-1}, ^nC_n$.The value of the binomial coefficients

Vedclass · Accepted Answer

$\frac{n}{2}$

Vedclass · Answer

(A) The binomial coefficient series is given by $^nC_0, ^nC_1, ^nC_2, \ldots, ^nC_r, \ldots, ^nC_{n-1}, ^nC_n$.The value of the binomial coefficients increases initially,reaches a maximum,and then decreases.For a given $n$,the total number of terms in the expansion is $n+1$.If $n$ is even,the total number of terms $n+1$ is odd,so there is a single middle term.The position of the middle term is given by $\frac{(n+1)+1}{2} = \frac{n}{2} + 1$.Since $^nC_r$ represents the $(r+1)$-th term,we set $r+1 = \frac{n}{2} + 1$,which gives $r = \frac{n}{2}$.Thus,when $n$ is even,the maximum value of $^nC_r$ occurs at $r = \frac{n}{2}$.

If $n$ is even and the value of $^nC_r$ is maximum,then $r = $

Similar Questions

In how many ways can a committee of $5$ members be formed from $6$ men and $4$ women such that the committee contains at least one woman?

In how many ways can a committee of $5$ members be formed from $4$ gentlemen and $6$ ladies such that the number of gentlemen is greater than the number of ladies?

In how many ways can $11$ identical pencils be distributed among $6$ children such that each child receives at least one pencil?

$^{20}C_1 + 3 ^{20}C_2 + 3 ^{20}C_3 + ^{20}C_4$ is equal to-

For a scholarship,at most $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selecting at least one candidate for the scholarship is $63$,then the maximum number of candidates that can be selected for the scholarship is -

Vedclass Test Series

Exam Paper Generator

Online Exam Module