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  4. Number of different terms in the sum (1+x)2009+(1+x2)2008+(1+x3)2007, is

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Q. Number of different terms in the sum $(1+x)^{2009}+\left(1+x^2\right)^{2008}+\left(1+x^3\right)^{2007}$, is

Binomial Theorem

Solution:

Number of terms in $(1+x)^{2009}=2010$....(1)
$+ \text { additional terms in }\left(1+x^2\right)^{2008}=x^{2010}+x^{2012}+\ldots \ldots .+x^{4016}=1004$...(2)
$+ \text { additional terms in }\left(1+x^3\right)^{2007}=x^{2010}+x^{2013}+\ldots \ldots . .+x^{4014}+\ldots .+x^{6021}=1338$...(3)
$\left.-(\text { common to } 2 \text { and } 3)=x^{2010}+x^{2016}+\ldots . . .+x^{4014}\right)=335$
$\text { Hence total }=2010+1004+1338-335 $
$=4352-335=4017 $