{What is } C(n, 1) + C(n, 2) + + C(n, n) { equal to}

Correct Answer is : 1+2+22+23+⋯+2n−11 + 2 + 2² + 2³ + + 2ⁿ-1}1+2+22+23+⋯+2n−1

Correct Answer is : 1+2+22+23+⋯+2n−11 + 2 + 2² + 2³ + + 2ⁿ-1}1+2+22+23+⋯+2n−1

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UPSC NDA I 2021 Mathematics Solved Paper

Question : 1 of 120

Marks: +1, -0

\text{What is } C(n, 1) + C(n, 2) + \dots + C(n, n) \text{ equal to}

Solution:

Concept:

The sum of binomial coefficients from

C(n,1)

C(n,n)

equals

2^n - 1

, which can be expressed as a geometric series.

Explanation:

Using the binomial theorem:

(1+1)^n = C(n,0) + C(n,1) + C(n,2) + \dots + C(n,n)

This simplifies to

2^n = C(n,0) + \left[C(n,1) + C(n,2) + \dots + C(n,n)\right]

Since

C(n,0) = 1

, we get

C(n,1) + C(n,2) + \dots + C(n,n) = 2^n - 1

Now,

2^n - 1

is a geometric series: first term

a = 1

, common ratio

r = 2

, number of terms

n

Sum of this series is

1 \cdot \frac{2^n - 1}{2 - 1} = 1 + 2 + 2^2 + \dots + 2^{n-1}

Thus, the required sum equals

1 + 2 + 2^2 + \dots + 2^{n-1}

Answer:

Option C:

1 + 2 + 2^{2} + \dots + 2^{n-1}

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