2021 AMC 12A Fall Problem 12

Question

What is the number of terms with rational coefficients among the

1001

terms in the expansion of

\left(x\sqrt[3]{2} + y\sqrt{3}\right)^{1000}?

0

166

167

500

501

The term for index

k

carries

2^{\frac{1000-k}{3}}\,3^{\frac{k}{2}};

rationality needs

1000 - k

to be a multiple of

3

and

k

to be a multiple of

2

These force

k \equiv 1 \pmod 3

and

k

even, i.e.

k \equiv 4 \pmod 6

The general term is

\binom{1000}{k}(x\sqrt[3]{2})^{1000-k}(y\sqrt{3})^{k},

whose coefficient contains

2^{\frac{1000-k}{3}}

and

3^{\frac{k}{2}}.

This is rational exactly when

(1000 - k)

is a multiple of

3

and

k

is even.

Since

1000 \equiv 1 \pmod 3,

we need

k \equiv 1 \pmod 3

and

k

even, which combine to

k \equiv 4 \pmod 6.

The valid values

k = 4, 10, \ldots, 1000

number

\dfrac{1000 - 4}{6} + 1 = 167.

Thus, the correct answer is C.

Problem 12 in other years