(2) If the coefficients of (2r + 4)th, (r - 2)th terms in the expansion of `(1 + x)^18` are equal, find r.

Explanation

In the expansion of `(1+x)^(18), " we have "T_(P+1)=.^(18)C_(p)x^(p)`.
`:. T_(2r+4)=T_((2r+3))+1=.^(18)C_((2r+3)) *x^(2r+3)" "` …(i)
and `T_((r-2))=T_((r-3)+1)=.^(18)C_(r-3)*x^(r-3)" "`…(ii)
It being given that the coefficients of `T_((2r+1))and T_((r-2))` are equal. So, from (i) and (ii), we get
`.^(18)C_(2r+3)=.^(18)C_((r-3)) rArr(2r-3)=(r-3)or(2r+3) +(r-3)=18`
`[:'.^(n)C_(p)=.^(n)C_(q) rArrp=q or p+q=n.]`
Now,`(2r+3)=(r-3) or (2r+3)+(r-3)=18`
`rArr=-6 or r=6`
`rArr r=6" " [ :' " the negative value of r is not permissible "]` .
Hence, r=6.