Let `S_(1)=underset(0 le i lt j le 100)(sumsum)C_(i)C_(j)`, `S_(2)=underset(0 le j lt i le 100)(sumsum)C_(i)C_(j)` and `S_(3)=underset(0 le i = j le 100)(sumsum)C_(i)C_(j)` where `C_(r )` represents cofficient of `x^(r )` in the binomial expansion of `(1+x)^(100)`
If `S_(1)+S_(2)+S_(3)=a^(b)` where `a`, `b in N`, then the least value of `(a+b)` is

Options

A. `66`
B. `72`
C. `46`
D. `52`

Answer

A

Explanation

`(a)` We have `S_(1)+S_(2)+S_(3)=sum_(i=0)^(100)sum_(j=0)^(100)C_(i)C_(j)`
`=('^(100)C_(0)+^(100)C_(1)+^(100)C_(2)+...+^(100)C_(100))^(2)`
`=(2^(100))^(2)=2^(200)`
Now `S_(1)+S_(2)+S_(3)=2^(200)=4^(10)=16^(50)=32^(40)=256^(25)=…=a^(b)`
Hence least value of `(a+b)=16+50=66`