Binomial Problem - 29 Dec

Question

!!sum_(r=0)\^n ((-1)^r)/(nC_r)!!

when !!n!! is an odd integer

!!a)" "0!!

!!b)" "n!!

!!c)" "(n(n+1))/2!!

!!d)" "!!None

Solution

Expanding given equation !!sum_(r=0)\^n ((-1)^r)/(nC_r)!!, we will get

!!1/(nC_0) - 1/(nC_1) + 1/(nC_2) - 1/(nC_3) ..........+ 1/(nC_(n-1)) - 1/(nC_n)!!

We know that !!nC_1 = nC_(n-1)!!

Using same concept in above equation sum of first term and last term will become zero.

Similarly for other terms as well.

Hence, !!sum_(r=0)\^n ((-1)^r)/(nC_r) = 0!!