Find the Coefficient

Probability Level 3

Find the coefficient of x 4 x^4 in the expansion of ( 2 − x + 3 x 2 ) 6 . \large (2-x+3x^2)^6.


The answer is 3660.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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2 solutions

Jakub Šafin
Nov 25, 2014

The full polynomial is ∑ k , l ( 6 k , l ) 2 6 − k − l ( − 1 ) k 3 l x k + 2 l , \sum_{k,l}{{6\choose k,l}2^{6-k-l}(-1)^k3^lx^{k+2l}}, from which we can see that the only summands that contain x 4 x^4 are for k = 0 , l = 2 ; k = 2 , l = 1 ; k = 4 , l = 0 k=0,l=2; k=2, l=1; k=4, l=0 . The answer is ( 6 0 , 2 ) 2 4 3 2 + ( 6 2 , 1 ) 2 3 3 + ( 6 4 , 0 ) 2 2 = 3660. {6\choose 0,2}2^43^2+{6\choose 2,1}2^33+{6\choose 4,0}2^2=3660.

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Neat solution! Did it same way! :)

Pranjal Jain - 6 years, 6 months ago
Durgesh Tiwari
Dec 1, 2014

(2+3x^2-x)^6 = 64(1+x{3x-1}/2)^6

LET U= x{3x-1}/2

(1+U)^6= 1+6U+15U^2+20U^3+15U^4+6U^5+U^6

EXTRACT THE TERMS OF x^4

coefficient of x^4= 64*(135/4 + 180/8 + 15/16) = 3660

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