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Question
If n is a positive integer and r is a non-negative integer, prove that the coefficients of xr and xn−r in the expansion of (1 + x)n are equal
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Solution
Given (1 + x)n.
General term tr+1 = nCr xn–r . ar
∴ The general term in the expansion of (1 + x)n is
Tr+1 = nCr . (1)n– r . xr
Tr+1 = nCr . xr ......(1)
∴ Coefficient of xr is nCr,
Put r = n – r in (1)
Tn–r+1 = nCn–r . xn–r
∴ The coefficient of xn-r is nCn-r ......(2)
We know nCr = nCn-r
∴ The coefficient of xr and coefficient of xn-r are equal, (by (1) and (2))
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