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प्रश्न
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 ______.
विकल्प
1
2
`1/2`
`1/"n"`
उत्तर
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 xn = 2nCn = A .....(Given)
In the expression (1 + x)2n–1
`"T"_(r + 1) = ""^(2n - 1)"C"_rx^r`
∴ Coefficient of xn = `""^(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]
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