Painful Multinomial?

Probability Level 2

What is the coefficient of x 15 x^{15} in

( 1 + 3 x + 3 x 2 + x 3 ) 10 ( 1 + 3x + 3x^2 + x^3 ) ^ {10}

( 30 10 ) 30 \choose 10 ( 15 10 ) { 15 \choose 10 } ( 30 15 ) { 30 \choose 15 } ( 25 15 ) { 25 \choose 15 }

Chung Kevin by Chung Kevin , 46, Australia

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1 solution

Chew-Seong Cheong
Oct 23, 2016

( 1 + 3 x + 3 x 2 + x 3 ) 10 = ( 1 + x ) 30 = ∑ n = 0 30 ( 30 n ) x n \displaystyle (1+3x+3x^2 + x^3)^{10} = (1+x)^{30} = \sum_{n=0}^{30} {30 \choose n}x^n , Therefore, the coefficient of x 15 x^{15} is ( 30 15 ) \boxed{\displaystyle {30 \choose 15}} .

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