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Question

The coefficient of middle term in the expansion of (1+x)40 is

A
1.3.53920!.220
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B
1.3.53920!
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C
4020!
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D
40! 2020
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Solution

The correct option is A 1.3.53920!.220
Middle term =N2+1
=2n2+1
=(n+1)thterm
Now consider the following
Tr+1=nCranrbr...(i)
Where T represents the term
Here the middle term is the (n+1)th term.
Hence, r+1=n+1
r=n
Substituting in (i), we get
2nCn12nnxn
=2nCnxn
Therefore the coefficient of the middle term is 2nCn
Substituting n=20, we get
40C20
=40!20!20!
Dividing by 20! and separating the multiples of two in the powers of two we get,
Middle term =(1.3.5.7....39)22020!

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