If C_0+C_1+C_2+ +C_n=256, then ^ 2 n C_2 is equal to: - Vidyadip

MCQ

If $C_0+C_1+C_2+\ldots+C_n=256$, then ${ }^{2 n} C_2$ is equal to:

A
56
✓
120
C
28
D
91

✓

Answer

Correct option: B.

120

120

Solution:
If set S has n elements, then C (n, k)C n, k is the number of ways of choosing k elements from S.
Thus, the number of subsets of SS of all possible values is given by,
$\text{C}(\text{n},0)+\text{C}(\text{n},1)+\text{C}(\text{n},3)+.....+\text{C}(\text{n},\text{n})=2^\text{n}$
Comparing the given equation with the above equation:
$2^\text{n}=256$
$\Rightarrow 2^\text{n}=2^{8}$
$\Rightarrow \text{n}=8$
$\therefore {^\text{2n}}\text{C}_{\text{2}}={^\text{16}}\text{C}_{\text{2}}$
$\Rightarrow {^\text{16}}\text{C}_{\text{2}}=\frac{16!}{2!4!}=\frac{16\times15}{2}=120$

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