Binomial Theorem And Its Simple Applications Question 16
Question: The coefficient of $ x^{n} $ in the polynomial $ (x+{{,}^{n}}C_0)(x+3.{{,}^{n}}C_1)(x+5.{{,}^{n}}C_2)…(x+{{(2n+1)}^{n}}C_{n}) $ is
Options:
A) $ n{{.2}^{n}} $
B) $ ~n{{.2}^{n+1}} $
C) $ (n+1){{.2}^{n}} $
D) $ n{{.2}^{n}}+1 $
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] $ (x+{{,}^{n}}C_0)(x+3.{{,}^{n}}C_1)(x+5.{{,}^{n}}C_2)…. $ $ (x+(2n+1).{{,}^{n}}C_{n}) $
$ ={x^{n+1}}+x^{n}\ ^{n}C_0+3. ^{n}C_1+5. ^{n}C_2+…. $ $ +(2n+1). ^{n}C_{n}+…… $
Coeff. of $ x^{n} $ $ ={{,}^{n}}C_0+3.{{,}^{n}}C_1+5.{{,}^{n}}C_2+….+(2n+1).{{,}^{n}}C_{n} $
$ =1+({{,}^{n}}C_1+2.{{,}^{n}}C_1)+({{,}^{n}}C_2+4.{{,}^{n}}C_2)+… $ $ +{{(}^{n}}C_{n}+2n.{{,}^{n}}C_{n}) $
$ =(1+{{,}^{n}}C_1+…+{{,}^{n}}C_{n})+2({{,}^{n}}C_1+2{{,}^{n}}C_2+…+n.{{,}^{n}}C_{n}) $
$ =2^{n}+2[ n+2.\frac{n(n-1)}{2!}+3.\frac{n(n-1)(n-2)}{3!}+…+n.1 ] $
$ =2^{n}+2n[1+{{,}^{n-1}}C_1+{{,}^{n-1}}C_2+….+{{,}^{n-1}}{C_{n-1}}] $
$ =2^{n}+2n{{.2}^{n-1}}=2^{n}(1+n)=(n+1){{.2}^{n}} $
BETA
