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Question: In the expansion of \[{\left( {1 + x} \right)^n}\], the sum of the coefficients of odd powers of \[x...

In the expansion of (1+x)n{\left( {1 + x} \right)^n}, the sum of the coefficients of odd powers of xx is?
A. 2n+1{2^n} + 1
B. 2n1{2^n} - 1
C. 2n{2^n}
D. 2n1{2^{n - 1}}

Explanation

Solution

We have to note down the given expression and expand it using the binomial expansion. Eliminate all the even terms out by adding or subtracting a certain constant which suits. Then solve it and simplify it to get the final answer.

Complete step-by-step solution:
Given expression,
(1+x)n{\left( {1 + x} \right)^n}
The expression (1+x)n{\left( {1 + x} \right)^n} can be expanded using the binomial theorem.
(1+x)n=nC0+nC1x+nC2x2+nC3x3+...+nCnxn{\left( {1 + x} \right)^n} = {}^n{C_0} + {}^n{C_1}x + {}^n{C_2}{x^2} + {}^n{C_3}{x^3} + ... + {}^n{C_n}{x^n}
Now, substituting the value of xx with x=1x = 1 and on the right hand side, subtracting 11 to eliminate the even values of the expression, we get;
(1+1)n=2(nC1+nC3+...+nCn){\left( {1 + 1} \right)^n} = 2\left( {{}^n{C_1} + {}^n{C_3} + ... + {}^n{C_n}} \right)
Adding the left-hand side and bringing the numeric value from the right-hand side to the left-hand side, we get;
2n2=(nC1+nC3+...+nCn)\dfrac{{{2^n}}}{2} = \left( {{}^n{C_1} + {}^n{C_3} + ... + {}^n{C_n}} \right)
Now, using the formula of exponents, we get;
nC1+nC2+nC5+...=2n1{}^n{C_1} + {}^n{C_2} + {}^n{C_5} + ... = {2^{n - 1}}

The correct option is D.

Note: The expansion of (1+x)n{\left( {1 + x} \right)^n} is given as the above by following the derivation given below;
(1+x)n{\left( {1 + x} \right)^n} can be written as
1+n1!x+n(n1)2!x2+n(n1)(n2)3!x3+...1 + \dfrac{n}{{1!}}x + \dfrac{{n\left( {n - 1} \right)}}{{2!}}{x^2} + \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{{3!}}{x^3} + ...
The expression is true for all the real values of nn although there are no conditions on xx.
If nn is a positive integer, then the expansion is terminated, but if nn is a negative integer or not an integer or both a combination of an integer and any other subject value, we have an infinite series which is only valid when x<1\left| x \right| < 1.
In algebra, especially in permutations and combinations, the binomial theorem describes the algebraic expansion of powers of a binomial, i.e., an expression containing two elements. It can be a polynomial, which is taken in terms of binomial by merging two elements or more into a single element and then applying the same binomial theorem to the expression.