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Number Theory Level 4

What is the coefficient of x 10 x^{10} in the expansion of ( 1 + x + x 2 + ⋯ + x 10 ) 10 \left(1+x+x^2+\dots+x^{10}\right)^{10} ?


The answer is 92378.

Ryan Phua by Ryan Phua , 22, Singapore

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1 solution

Ryan Phua
Dec 24, 2015

The coefficient of x 10 x^{10} is equal to the addition of exponents of 1 term from each of the 10 brackets of ( 1 + x + x 2 + … + x 10 ) (1+x+x^2+ \ldots + x^{10}) . Thus, the problem could be viewed as finding the number of non-negative solutions to the equation, a + b + c + … + h + i + j = 10 a+b+c+\ldots+h+i+j = 10 . By stars & bars, the number of solutions is therefore ( 19 9 ) = 92378 {19 \choose 9} = \boxed{92378} .

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It is also the coefficient of the x 10 x^{10} term in the binomial expansion of ( 1 − x ) − 10 (1-x)^{-10} .

John Frank - 3 years, 9 months ago

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