Question:
The coefficient of $x^n$ in the polynomial
$\left(x+^{n}C_{0}\right)\left(x+3. ^{n}C_{1}\right)\left(x+5. ^{n}C_{2}\right) .... \left(x+\left(2n+1\right)^{n}C_{n}\right)$ is
The coefficient of $x^n$ in the polynomial
$\left(x+^{n}C_{0}\right)\left(x+3. ^{n}C_{1}\right)\left(x+5. ^{n}C_{2}\right) .... \left(x+\left(2n+1\right)^{n}C_{n}\right)$ is
Updated On: Mar 2, 2023
- $n . 2^n$
- $n . 2^{n + 1}$
- $(n + 1) . 2^n$
- $(n + 1) . 2^{n+1}$
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
$\left(x+^{n}C_{0}\right)\left(x+3. ^{n}C_{1}\right)\left(x+5. ^{n}C_{2}\right) .... \left(x+\left(2n+1\right).^{n}C_{n}\right)$
$= x^{n+1}+x^{n}\left\{^{n}C_{0}+3. ^{n}C_{1}+5. ^{n}C_{2}+..... +\left(2n+1\right).^{n}C_{n}\right\}+.....$
Coeff. of $x^{n} = ^{n}C_{0} +3. ^{n}C_{1}+5. ^{n}C_{2}+..... +\left(2n+1\right).^{n}C_{n}$
$ = 1+ \left(^{n}C_{1} +2. ^{n}C_{1}\right)+\left(^{n}C_{2}+4. ^{n}C_{2}\right) + ....+\left(^{n}C_{n}+2n. ^{n}C_{n}\right)$
$= \left(1+^{n}C_{1}+..... + ^{n}C_{n}\right)+2\left(^{n}C_{1} + 2^{n}C_{2} + .... +n. ^{n}C_{n}\right)$
$= 2^{n} +2\left[n+2. \frac{n\left(n-1\right)}{2!}+3. \frac{n\left(n-1\right)\left(n-2\right)}{3!}+...+ n.1\right]$
$= 2^{n} +2n \left[1+^{n-1}C_{1}+^{n-1}C_{2}+ ..... +^{n-1}C_{n-1}\right]$
$= 2^{n}+2^{n}. 2^{n-1} = 2^{n} \left(1+n\right) = \left(n+1\right) . 2^{n}$
Was this answer helpful?
1
0
Top Questions on binomial expansion formula
- Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
- JEE Advanced - 2025
- Mathematics
- binomial expansion formula
- For some \( n \neq 10 \), let the coefficients of the 5th, 6th, and 7th terms in the binomial expansion of \( (1 + x)^{n+4} \) be in A.P. Then the largest coefficient in the expansion of \( (1 + x)^{n+4} \) is:
- JEE Main - 2025
- Mathematics
- binomial expansion formula
- Let \( a_n \) denote the term independent of \( x \) in the expansion of \( \left[ x + \frac{\sin(1/n)}{x^2} \right]^{3n} \). Then \( \lim_{n \to \infty} \frac{(a_n) n!}{3^n n^n} \) equals:
- WBJEE - 2025
- Mathematics
- binomial expansion formula
- Find the term independent of $ x $ in $ \left( 2x - \frac{5}{x^2} \right)^6 $
- KEAM - 2025
- Mathematics
- binomial expansion formula
- Binomial Expansion Series $ \left( \frac{(1 + x)}{(n + 1)} \right)' = n_0 x + n_1 \frac{x^2}{2} + n_2 \frac{x^3}{3} + \cdots + n_n \frac{x^n}{n+1} $
- MHT CET - 2025
- Mathematics
- binomial expansion formula
View More Questions
Notes on binomial expansion formula
Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .