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Find the coefficient of t^8 in the expa...

Find the coefficient of `t^8` in the expansion of `(1+2t^2-t^3)^9`.

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We have `((1+2t^(2))-t^(3))^(9)=""^(9)C_(0)(1+2t^(2))^(9)-""^(9)C_(1)(1+2t^(2))^(8)*t^(3)+""^(9)C_(2)(1+2t^(2))^(7)*t^(6)-""^(9)C_(3)(1+2t^(2))^(6)*t^(9)+…….-""^(9)C_(9)(t^(3))^(9)`
`:.` Coefficient of `t^(8)` in the expansion of `(1+2t^(2)-t^(3))^(9)`
`=""^(9)C_(0)`(coefficient of `t^(8)` in `(1+2t^(2))^(9)`)-`""^(9)C_(1)`(coefficient of `t^(5)` in `(1+2t^(2))^(8)`)+`""^(9)C_(2)` ( coefficient of `t^(3)` in `(1+2t^(2))^(7)`)
`=""^(9)C_(0)*"'^(9)C_(4) 2^(4)=0+""^(9)C_(2)*""^(7)C_(1)*2=(9*8*7*6)/(4*3*2*1)*16+(9.8)/(2.1)*7*2=9*8*7*8=2520`
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