Huge Bionomial

Algebra Level 4

Find the coefficient of x 9 x^9

( 2 + x ) 10 + ( 2 + x ) 9 ( 1 + x ) + ( 2 + x ) 8 ( 1 + x ) 2 + . . . + ( 1 + x ) 10 (2+x)^{10} + (2+x)^9(1+x)+(2+x)^8(1+x)^2+...+(1+x)^{10}


The answer is 165.

Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Aug 21, 2015

( 2 + x ) 10 + ( 2 + x ) 9 ( 1 + x ) + ( 2 + x ) 8 ( 1 + x ) 2 + . . . + ( 1 + x ) 10 (2+x)^{10} + (2+x)^9(1+x)+(2+x)^8(1+x)^2+...+(1+x)^{10} = ( 2 + x ) 11 − ( 1 + x ) 11 ( 2 + x ) − ( 1 + x ) = ( 2 + x ) 11 − ( 1 + x ) 11 = \frac{(2+x)^{11} - (1+x)^{11}}{(2+x) - (1+x)} = (2+x)^{11} - (1+x)^{11}

This implies that the x 9 x^9 term coefficient is ( 11 9 ) 2 2 − ( 11 9 ) = 165 {11 \choose 9}2^2 - {11 \choose 9} = \boxed{165}

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