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Middle Terms

If 'p' and 'q...

Question

If 'p' and 'q' are the coefficients of $x^{a}$ and $x^{b}$ respectively in $(1 + x)^{a + b}$ , then

2 p = q

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p + q = 0

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p = q

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p = 2 q

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Solution

The correct option is C $p = q$
Given $(1 + x)^{a + b}$
General term
$T_{r + 1} =^{n} C_{r} x^{n - r} y^{r}$
$T_{r + 1} =^{n} C_{r} x^{n - r} y^{r}$
$T_{r + 1} =^{a + b} C_{r} (1)^{a + b - r} (x)^{r}$
given 'p' the coefficeint of $x^{a}$
$r = a$
$T_{r + 1} =^{a + b} C_{a} (1)^{a + b - a} (x)^{a}$
$T_{r + 1} =^{a + b} C_{a} (1)^{b} (x)^{a}$
given 'q' the coefficient of $x^{b}$
$r = b$
$T_{r + 1} =^{a + b} C_{b} (1)^{a + b - b} (x)^{b}$
$T_{r + 1} =^{a + b} C_{b} (1)^{a} (x)^{b}$
$p =^{a + b} C_{a}$
$q =^{a + b} C_{b} =^{a + b} C_{a + b - b} =^{a + b} C_{a}$
Using $^{n} C_{r} =^{n} C_{n - r}$
Hence $p = q$ .

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Similar questions

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If 'p' and 'q' are the coefficients of xa and xb respectively in (1+x)a+b, then

If 'p' and 'q' are the coefficients of $x^{a}$ and $x^{b}$ respectively in $(1 + x)^{a + b}$ , then