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Question

If the coefficients of (2r+4)th and (r2)th terms in the expansion of (1+x)18 are equal, then the value of r is

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Solution

In the expansion of (1+x)18, we have Tp+1= 18Cpxp
T(2r+4)=T(2r+3)+1= 18C2r+3.x2r+3(1)
and T(r2)=T(r3)+1= 18C(r3).xr3(2)
It is given that the coefficients of T(2r+4) and T(r2) are equal.
So, from (1) and (2), we get
18C2r+3= 18Cr3
(2r+3)=(r3) or (2r+3)+(r3)=18 [ nCp=nCqp=q or p+q=n]
Now, (2r+3)=(r3) or (2r+3)+(r3)=18
r=6 or r=6
r=6
[ the negative value of r is not permissible].
Hence, r=6

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