Question

Sum of the binomial coefficients is​

Answer 1

Answer:

binomial coefficients: If α is a nonnegative integer n, then all terms with k > n are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However, for other values of α, including negative integers and rational numbers, the series is really infinite.

Answer 2

Step-by-step explanation:

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Study Material»IIT JEE Mathematics»Algebra»Binomial Theorem for a Positive Integral Index»Sum of Binomial Coefficients

Sum of Binomial Coefficients

Putting x = 1 in the expansion (1+x)n = nC0 + nC1 x + nC2 x2 +...+ nCx xn, we get,

2n = nC0 + nC1 x + nC2 +...+ nCn.

We kept x = 1, and got the desired result i.e. ∑nr=0 Cr = 2n.

Note: This one is very simple illustration of how we put some value of x and get the solution of the problem. It is very important how judiciously you exploit this property of binomial expansion.

Illustration:

Find the value of C0 + C2 + C4 +... in the expansion of (1+x)n.

Solution:

We have,

(1 + x)n = nC0 + nC1 x + nC2 x2 +... nCn xn.

Now put x = -x; (1-x)n = nC0 - nC1 x + nC2 x2 -...+ (-1)n nCn xn.

Now, adding both expansions, we get,

(1 + x)n + (1-x)n = 2[nC0 + nC2 x2 + nC4 x4 +.........]

Put x = 1

=> (2n+0)/2 = C0 + C2 + C4 +......

or C0 + C2 + C4 +......= 2n-1

Explanation:

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