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