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### AIEEE 2002

The coefficients of $${x^p}$$ and $${x^q}$$ in the expansion of $${\left( {1 + x} \right)^{p + q}}$$ are
A
equal
B
equal with opposite signs
C
reciprocals of each other
D
none of these

## Explanation

Here in this expansion $${\left( {1 + x} \right)^{p + q}}$$

The general term = $${T_{r + 1}} = {}^{p + q}{C_r}.{\left( x \right)^r}$$

$$\therefore$$ $${x^p}$$ will be present in the term = $${}^{p + q}{C_p}.{\left( x \right)^p}$$

So coefficient of $${x^p}$$ = $${}^{p + q}{C_p}$$

And $${x^q}$$ will be present in the term = $${}^{p + q}{C_q}.{\left( x \right)^q}$$

$$\therefore$$ coefficient of $${x^q}$$ = $${}^{p + q}{C_q}$$

We know $${}^n{C_r}$$ = $${}^n{C_{n - r}}$$

$$\therefore$$ $${}^{p + q}{C_q}$$ = $${}^{p + q}{C_{\left( {p + q} \right) - q}}$$ = $${}^{p + q}{C_p}$$

So coefficients of $${x^p}$$ and $${x^q}$$ are equal.

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