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57 votes
57 votes
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.
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17 Answers

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7 votes

= [ x( 1 + x + x+ x+ ....................... ) ]3

= x( 1 + x + x+ x+.......................)3

= x(1 / ( 1 - x ) )3

= x( 1 / ( 1 - x ))

= x( ( 1 - x )-3 )

= x( 1 + (3)x + ( (3*4) / (1*2) ) x+ ( (3*4*5) / (1*2*3) ) x+...................................)

= x+ 3x10 + 6x11 + 10x12 +........................................................

Answer = 10

4 votes
4 votes

[x12](x3 + x4 + x5 +x6 +...)3 = [x3](1 + x + x2 + x3 +x4 +...)= [x3]((1-x)-1)3 =  [x3](1-x)-3

 = -3C3(-1)3 = 5C3(-1)3(-1)=  5C= 10

Answer:

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