JEE Main Solved Paper 2017 Question 19
Question: The value of $ ({{ }^{21}}C _1-{{ }^{10}}C _1)+({{ }^{21}}C _2-{{ }^{10}}C _2)+ $ $ ({{ }^{21}}C _3-{{ }^{10}}C _3)+({{ }^{21}}C _4-{{ }^{10}}C _4)+….+ $ $ ({{ }^{21}}C _{10}-{{ }^{10}}C _{10}) $ is:- [JEE Main Solved Paper-2017]
Options:
A) $ 2^{20}-2^{10} $
B) $ 2^{21}-2^{11} $
C) $ 2^{21}-2^{10} $
D) $ 2^{20}-2^{9} $
Show Answer
Answer:
Correct Answer: A
Solution:
$ ({{}^{21}}C _1{{+}^{21}}C _2…..+{{ }^{21}}C _{10}) $ $ -({{}^{10}}C _1{{+}^{10}}C _2…..{{ }^{10}}C _{10}) $
$ =\frac{1}{2}[({{ }^{21}}C _1+….+{{ }^{21}}C _{10})+({{ }^{21}}C _{11}+….{{ }^{21}}C _{20})] $ $ -(2^{10}-1) $
$ =\frac{1}{2}[2^{21}-2]-(2^{10}-1) $
$ =(2^{20}-1)-(2^{10}-1)=2^{20}-2^{10} $
