Find coefficient of x⁵ in expansion of (1+x)¹⁰(1-x)¹⁰

Question

Find the coefficient of $x^5$ in the expansion of $(1+x)^{10}(1-x)^{10}$ .

(JEE Advanced 2023, similar pattern)

Solution — Step by Step

Notice that $(1+x)^{10}(1-x)^{10} = [(1+x)(1-x)]^{10} = (1-x^2)^{10}$ .

This is a crucial simplification — instead of multiplying two 11-term expansions, we now work with a single binomial.

(1 - x^2)^{10} = \sum_{r=0}^{10} \binom{10}{r}(-x^2)^r = \sum_{r=0}^{10} \binom{10}{r}(-1)^r x^{2r}

The general term has $x^{2r}$ — only EVEN powers of $x$ appear.

We need $2r = 5$ , which gives $r = 5/2$ . Since $r$ must be a non-negative integer, there is no term with $x^5$ .

\text{Coefficient of } x^5 = \mathbf{0}

Why This Works

The product $(1+x)(1-x) = 1 - x^2$ eliminates the odd powers of $x$ . When raised to the 10th power, $(1-x^2)^{10}$ contains only $x^0, x^2, x^4, x^6, \ldots, x^{20}$ — all even powers. No odd power of $x$ can ever appear.

This is a symmetry argument: $(1+x)^{10}(1-x)^{10}$ is an even function (replacing $x$ with $-x$ gives the same expression), so its expansion has only even-powered terms.

Alternative Method

Without the simplification, you would need to convolve the two expansions. The coefficient of $x^5$ in $(1+x)^{10}(1-x)^{10}$ is:

\sum_{k=0}^{5} \binom{10}{k} \cdot \binom{10}{5-k}(-1)^{5-k}

= \sum_{k=0}^{5} (-1)^{5-k}\binom{10}{k}\binom{10}{5-k}

You can verify this sum equals 0 by pairing terms: $k$ and $5-k$ produce terms that cancel pairwise. But the simplification to $(1-x^2)^{10}$ makes the answer obvious in one step.

Whenever you see $(1+x)^n(1-x)^n$ or similar products, always check if the product simplifies. $(1+x)(1-x) = 1-x^2$ , $(1+x^2)(1-x^2) = 1-x^4$ , etc. This is the single most useful trick in binomial coefficient problems.

Common Mistake

Students often jump into expanding both binomials separately and then collecting the $x^5$ terms — a tedious and error-prone process with 6 terms to add. The clever approach is to simplify first. If you do not spot the $(1-x^2)^{10}$ simplification, you waste 5-7 minutes on a problem that should take 30 seconds. Always look for algebraic identities before expanding.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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