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MCQ

Fill in the Blanks

If A and B are coefficient of x n in the expansions of (1 + x)^{2n} and (1 + x)^{2n–1} respectively, then `A/B` equals ______.

#### Options

1

2

`1/2`

`1/"n"`

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#### Solution

If A and B are coefficient of x n in the expansions of (1 + x)^{2n} and (1 + x)^{2n–1} respectively, then `A/B` equals **2**.

**Explanation:**

Given expression is (1 + x)^{2n}

`"T"_(r + 1) = ""^(2n)"C"_r x^r`

∴ Coefficient of x^{n} = ^{2n}C_{n} = A .....(Given)

In the expression (1 + x)^{2n–1}

`"T"_(r + 1) = ""^(2n - 1)"C"_rx^r`

∴ Coefficient of x^{n} = `""^(2n - 1)"C"_n` = B ....(Given)

So, `A/B = (""^(2n)"C"_n)/(""^(2n - 1)"C"_n)`

= `(""^(2n)"C"_n)/(""^(2n - 1)"C"_n) = 2/1` ....[From Q. no 21]

Concept: Proof of Binomial Therom by Combination

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