Binomial Theorem And Its Simple Applications Question 30
Question: The coefficient of $ x^{100} $ in the expansion of $ \sum\limits_{j=0}^{200}{{{(1+x)}^{j}}} $ is:
Options:
A) $ {{,}^{200}}C_{100} $
B) $ {{,}^{201}}C_{102} $
C) $ {{,}^{200}}C_{101} $
D) $ {{,}^{201}}C_{100} $
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] $ {{(1+x)}^{n}}=1+{{,}^{n}}C_1x+{{,}^{n}}C_2x^{2}+{{,}^{n}}C_3x^{3}+….. $
$ {{+}^{j}}C_{100}x^{100}+……+{{+}^{j}}C_{200}x^{200} $
$ \therefore $ Coefficient of $ x^{100} $ in the expansion of $ {{(1+x)}^{j}}={{,}^{j}}C_{100} $ Coefficient of $ x^{100} $ in the expansion of $ \sum\limits_{j=0}^{200}{{{(1+x)}^{j}}} $ will be equal to $ \sum\limits_{j=100}^{200}{^{j}C_{100}} $
$ ={{,}^{100}}C_{100}+{{,}^{101}}C_{100}+{{,}^{102}}C_{100}+….+{{,}^{200}}C_{100} $
$ ={{,}^{200}}C_{100} $
BETA
