Binomial Theorem And Its Simple Applications Question 20
Question: Which of the following is the greatest-
Options:
A) $ ^{31}C_0^{2}-{{,}^{31}}C_1^{2}+{{,}^{31}}C_2^{2}-…-{{,}^{31}}C_31^{2} $
B) $ ^{32}C_0^{2}-{{,}^{32}}C_1^{2}+{{,}^{32}}C_1^{2}-…+{{,}^{32}}C_32^{2} $
C) $ ^{32}C_0^{2}+{{,}^{32}}C_1^{2}+{{,}^{32}}C_2^{2}-..+{{,}^{32}}C_32^{2} $
D) $ ^{34}C_0^{2}-{{,}^{34}}C_1^{2}+{{,}^{34}}C_2^{2}-…+{{,}^{34}}C_32^{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] We know that $ ^{n}C_0^{2}+{{,}^{n}}C_1^{2}+…+{{,}^{n}}C_n^{2}=2{{,}^{n}}C_{n} $ and $ ^{n}C_0^{2}-{{,}^{n}}C_1^{2}+…+{{,}^{n}}C_n^{2} $
$ \begin{aligned} & = \begin{cases} 0, \\ {C_{n/2}}{{(-1)}^{n/2}}, \\ \end{cases} .\begin{matrix} ifnisodd \\ fniseven \\ \end{matrix} \\ & , \\ & ^{,}i \\ \end{aligned} $
From this $ ^{31}C_0^{2}-{{,}^{31}}C_0^{2}+{{,}^{31}}C_2^{2}-…-{{,}^{31}}C_31^{2}=0 $
$ ^{32}C_0^{2}-{{,}^{32}}C_1^{2}+{{,}^{32}}C_2^{2}-…+{{,}^{32}}C_32^{2}=-{{,}^{32}}C_{16} $
$ ^{34}C_0^{2}-{{,}^{34}}C_1^{2}+{{,}^{34}}C_2^{2}-…+{{,}^{34}}C_32^{2}=-{{,}^{34}}C_{17} $
$ ^{32}C_0^{2}+{{,}^{32}}C_1^{2}+{{,}^{32}}C_2^{2}-…+{{,}^{32}}C_32^{2}={{,}^{64}}C_{32} $ Obviously $ ^{64}C_{32} $ is greatest.
BETA
