2018 AMC 12B Problem 22

Problem 22 of 25HarderAlgebra Combinatorics

Consider polynomials

P(x)

of degree at most

3,

each of whose coefficients is an element of

\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}.

How many such polynomials satisfy

P(-1)=-9?

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Solution

Write

P(x)=ax^3+bx^2+cx+d

with each of

a,b,c,d

\{0,\ldots,9\}.

The condition is

-a+b-c+d=-9.

Let

a'=9-a

and

c'=9-c,

both in

[0,9].

Then

a'+b+c'+d=9.

By stars and bars the number of nonnegative solutions is

\binom{9+3}{3}=\binom{12}{3}=220,

and each automatically satisfies the upper bounds since the sum is

9.

Thus, the correct answer is D.

Tagged: stars and bars · substitution