Mathematics - Binomial Theorem Question with Solution | TestHub

$(1+x{)}^{2009}$ has 2010 terms in total. ${\left(1+x^2\right)}^{2008}$ has a constant, even power of x starting from 2 to 4016 but already even powers of x from 2 to 2008 were enumerated in $(1+x{)}^{2009}$ . The remaining terms containing even powers of x are from 2010 to 4016. They are 1004 in number. In ${\left(1+x^3\right)}^{2007}$ has a constant, multiples of 3 as powers of x. Even multiples of 3 from 6 to 4014 were already enumerated in above expansions. The remaining even multiples of 3 from 4020 to 6018 are 334 in number. Odd multiples of 3 as powers of x from 3 to 2007 were enumerated in above expansions and the remaining from 2013 to 6021 are to be enumerated. They are 669 in number. ∴ the number of terms in the expansion $=2010+1004+669+334=4017$.