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The ratio of the coefficients of x^{p} and x^{q} in the expansion of (1 + x)^{p + q} is ______.

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#### Solution

The ratio of the coefficients of x^{p} and x^{q} in the expansion of (1 + x)^{p + q} is **1 : 1**.

**Explanation:**

Given expansion is (1 + x)^{p + q}

T_{r+1} = `""^(p + q)"C"_r x^r`

Put r = p = `""^(p + q)"C"_p x^p`

∴ The coefficient of x^{p} = `""^(p + q)"C"_p`

Similarly, coefficient of x^{q} = `""^(p + q)"C"_q`

`""^(p + q)"C"_p = ((p + q)!)/(p!(p + q - p)!)`

= `((p + q)!)/(p!q!)`

And `""^(p + q)"C"_q = ((p + q)!)/(q!(p + q - q)!)`

= `((p + q)!)/(p!q!)`

So, the ratio is 1 : 1.

Concept: Proof of Binomial Therom by Combination

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