# The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n–1 are in the ratio ______. - Mathematics

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The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n–1 are in the ratio ______.

• 1 : 2

• 1 : 3

• 3 : 1

• 2 : 1

#### Solution

The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n–1 are in the ratio 2 : 1

Explanation:

General Term "T"_(r + 1) = ""^"n""C"_r  x^(n - r) y^r

In the expansion of (1 + x)2n

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

To get the coefficient of xn

Put r = n

∴ Coefficient of xn = 2nCn

In the expansion of (1 + x)2n–1

We get "T"_(r + 1) = ""^(2n - 1)"C"_r x^r

∴ Coefficient of xn = ""^(2n - 1)"C"_(n - 1)

The required ratio is (""^(2n)"C"_n)/(""^(2n - 1)"C"_(n - 1))

= ((2n!)/(n!(n!)))/(((2n - 1)!)/((n - 1)!(2n - 1 - n + 1)!))

= ((22n!)/(n!n!))/(((2n - 1)!)/((n - 1)(n!))

= (2n!)/(n!n!) xx ((n - 1)!*n!)/((2n - 1)!)

= (2n(2n - 1)!)/(n!n(n - 1)!) xx ((n - 1)!*n!)/((2n - 1)!)

= 2/1

= 2 : 1

Concept: Proof of Binomial Therom by Combination
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Exercise | Q 21 | Page 144
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