1.

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

Answer»

`66`
`72`
`46`
`52`

Solution :`(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`


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