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In the expansion of ( 1 + x+ x^(2) + x^(...

In the expansion of `( 1 + x+ x^(2) + x^(3))^(6)` , the coefficient of `x^(14)` is `:`

Text Solution

Verified by Experts

The correct Answer is:
120
`(1+x+x^(2)+x^(3))^(6)=(1+x)^(6)(1+x^(2))^(6)`
`=(""^(6)C_(0)+""^(6)C_(1)x+""^(6)C_(2)x^(2)+""^(6)C_(3)x^(3)+""^(6)C_(4)x^(4)+""^(6)C_(5)x^(5)+""^(6)C_(6)x^(6))`
`xx(""^(6)C_(0)+""^(6)C_(1)^(2)+""^(6)C_(2)x^(4)+""^(6)C_(3)x^(6)""^(6)C_(4)x^(8)+""^(6)C_(5)x^(10)+""^(6)C_(6)x^(12))`
`therefore" Coefficient of "x^(14)" in "(1+x+x^(2)+x^(3))^(6)`
`=""^(6)C_(2).""^(6)C_(6)+""^(6)C_(4).""^(6)C_(5)+""^(6)C_(6).""^(6)C_(4)=15+90+15=120`
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