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Question
The ratio of the coefficients of xp and xq in the expansion of (1 + x)p + q is ______.
Solution
The ratio of the coefficients of xp and xq in the expansion of (1 + x)p + q is 1 : 1.
Explanation:
Given expansion is (1 + x)p + q
Tr+1 = `""^(p + q)"C"_r x^r`
Put r = p = `""^(p + q)"C"_p x^p`
∴ The coefficient of xp = `""^(p + q)"C"_p`
Similarly, coefficient of xq = `""^(p + q)"C"_q`
`""^(p + q)"C"_p = ((p + q)!)/(p!(p + q - p)!)`
= `((p + q)!)/(p!q!)`
And `""^(p + q)"C"_q = ((p + q)!)/(q!(p + q - q)!)`
= `((p + q)!)/(p!q!)`
So, the ratio is 1 : 1.
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