If C0 + C1 + C2 + ...Cn = 256, then 2nC2 is equal to :A. 56 B. 120 C

1.	If C0 + C1 + C2 + ...Cn = 256, then 2nC2 is equal to :A. 56 B. 120 C. 28 D. 91
Answer» Option : (B) If set S has n elements, then C(n,k) is the number of ways of choosing k elements from S. Thus, The number of subsets of S of all possible values is given by, C(n,0) + C(n,1) + C(n,2) + . . . . . . . . . . + C(n,n) = 2ⁿ Comparing the given equation with the above equation we get, 2ⁿ = 256 2ⁿ = 28 n = 8 ∴ ²ⁿC₂ = ¹⁶C₂ = \(\frac{16!}{2!14!}\) = \(\frac{16\times 15}{2}\) = 120

If C0 + C1 + C2 + ...Cn = 256, then 2nC2 is equal to :A. 56 B. 120 C. 28 D. 91

Answer»

Option : (B)

If set S has n elements, then C(n,k) is the number of ways of choosing k elements from S.

Thus,

The number of subsets of S of all possible values is given by,

C(n,0) + C(n,1) + C(n,2) + . . . . . . . . . . + C(n,n) = 2ⁿ

Comparing the given equation with the above equation we get,

2ⁿ = 256

2ⁿ = 28

n = 8

∴ ²ⁿC₂ = ¹⁶C₂

= \(\frac{16!}{2!14!}\)

= \(\frac{16\times 15}{2}\)

= 120