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  • The number of different terms in the expansion of  <img alt="(1+x)^{2009}+\left(1+x^2\right)^{2008}+\left(1+x^3\right)

The number of different terms in the expansion of  (1+x)^{2009}+\left(1+x^2\right)^{2008}+\left(1+x^3\right)^{2007} is

 

Option: 1

3683


Option: 2

4017


Option: 3

4018


Option: 4

4352


Answers (1)

best_answer

(1+\mathrm{x})^{2009} has 2010 terms in total.  \left(1+\mathrm{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 which 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.

\therefore the number of terms in the expansion =2010+1004+669+334=4017.

Posted by

sudhir.kumar

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