GATE CSE 2016 Set 1 | Question: 26

Question

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

Comments

Very good method @Sourav.Sharma

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The problem can read as :

You are given 3 buckets and you need to pick out 12 items, but each bucket only allows you to pick at least 3 items in one go (not less than that)

We can easily shift the origin from 3 to 0 (as 3 items are always compulsory to pick). So 9 will be picked anyway. We can only decide on the remaining 3.

And now we only need to pick 3 items (and we are allowed to pick any number of items starting from 1, 2,)

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Answer 1

= [ x³ ( 1 + x + x² + x³ + ....................... ) ]³

= x⁹ ( 1 + x + x² + x³ +.......................)³

= x⁹ (1 / ( 1 - x ) )³

= x⁹ ( 1 / ( 1 - x )³ )

= x⁹ ( ( 1 - x )^-3 )

= x⁹ ( 1 + (3)x + ( (3*4) / (1*2) ) x² + ( (3*4*5) / (1*2*3) ) x³ +...................................)

= x⁹ + 3x¹⁰ + 6x¹¹ + 10x¹² +........................................................

Answer = 10

Answer 2

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

 ^= -3C₃(-1)³ = ⁵C₃(-1)³(-1)³ =  ⁵C₃ = 10

Answer 3

best way of solving these type of question...

Answer 4

Answer will be 10

we will see why