Question:
The total number of terms in the expansion of (1 + x)2n – (1 – x)2n ___________.
Solution:
In $(1+x)^{2 n}-(1-x)^{2 n}$
Since
$(1+x)^{2 n}={ }^{2 n} C_{0}+{ }^{2 n} C_{1} x+\ldots \ldots+{ }^{2 n} C_{2 n} x^{2 n}$
$(1-x)^{2 n}={ }^{2 n} C_{0}-{ }^{2 n} C_{1} x+\ldots \ldots+{ }^{2 n} C_{2 n} x^{2 n}$
Subtracting above two,
i. e. $(1+x)^{2 n}-(1-x)^{2 n}=2\left[{ }^{2 n} C_{1} x+{ }^{2 n} C_{3} x^{3}+\ldots{ }^{2 n} C_{2 n-1} x^{2 n-1}\right]$
i. e. number of terms here $=2 \times \frac{n}{2}($ odd from $2 n$ is $n)$
$=n$
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