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Sum

Find the coefficient of x^{15} in the expansion of (x – x^{2})^{10}.

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#### Solution

The given expression is (x – x^{2})^{10}

General Term `"T"_(r + 1) = ""^n"C"_r ^(n - r) y^r`

= `""^10"C"_r (x)^(10 - r) (- x^2)r`

= `""^10"C"_r (x)^(10 - r) (-1)^r * (x^2)^r`

= `(-1)^r * ""^10"C"_r (x)^(10 - r + 2r)`

= `(-1)^r * ""^10"C"_r (x)^(10 + r)`

To find the coefficient of x^{15}

Put 10 + r = 15

⇒ r = 5

∴ Coefficient of x^{15} = `(-1)^5 ""^10"C"_5`

= `- ""^10"C"_5`

= – 252

Hence, the required coefficient = – 252

Concept: Binomial Theorem for Positive Integral Indices

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