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### AIEEE 2002

$$r$$ and $$n$$ are positive integers $$\,r > 1,\,n > 2$$ and coefficient of $$\,{\left( {r + 2} \right)^{th}}$$ term and $$3{r^{th}}$$ term in the expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal, then $$n$$ equals
A
$$3r$$
B
$$3r + 1$$
C
$$2r$$
D
$$2r + 1$$

## Explanation

$$\,{\left( {r + 2} \right)^{th}}$$ term = $${}^{2n}{C_{r+1}}{\left( x \right)^r}$$

And coefficient of $$\,{\left( {r + 2} \right)^{th}}$$ = $${}^{2n}{C_{r+1}}$$

$$3{r^{th}}$$ term = $${}^{2n}{C_{3r - 1}}{\left( x \right)^{3r - 1}}$$

And coefficient of $$3{r^{th}}$$ term = $${}^{2n}{C_{3r - 1}}$$

According to the question,
$${}^{2n}{C_{r+1}}$$ = $${}^{2n}{C_{3r - 1}}$$

$$\Rightarrow \left( {r + 1} \right) + \left( {3r - 1} \right) = 2n$$

[As if $${}^n{C_p} = {}^n{C_q}$$ then p + q = n]

$$\Rightarrow 4r = 2n$$

$$\Rightarrow n = 2r$$
2

### 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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