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

Option 1)
$3r\;$

Option 2)
$\; 3r+1\;$

Option 3)
$\; 2r\;$

Option 4)
$\; 2r+1$

Answers (1)

As we learnt in

Properties of Binomial Theorem -

If $^{n}c_{r}= ^{n}c_{s}$ then

$r= s \, or \, r+s= n$

-

In $(1+x)^{2n}$

Coefficient of $T_{r+2}=\ ^{2n}C_{r+1}$ and Coefficient of $T_{3r}=\ ^{2n}C_{3r-1}$

So if $\ ^{2n}C_{r+1}=\ ^{2n}C_{3r-1}$

Then either r + 1 = 3r - 1

i.e. r = 1

or r + 1 + 3r - 1 = 2n

$r=\frac{n}{2}\ \; \Rightarrow\ \;n = 2r$

Correct option is 3.

Option 1)

$3r\;$

This is an incorrect option.

Option 2)

$\; 3r+1\;$

This is an incorrect option.

Option 3)

$\; 2r\;$

This is an incorrect option.

Option 4)

$\; 2r+1$

This is an incorrect option.

Posted by

Sabhrant Ambastha

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