1958 AMC 12 第 49 题

Question

(a+b)^n

的展开式中有

n+1

个不同项。

(a+b+c)^{10}

的展开式中不同项的个数为：

In the expansion of

(a+b)^n

there are

n+1

dissimilar terms. The number of dissimilar terms in the expansion of

(a+b+c)^{10}

is:

11

33

55

66

132

每一项由和为

10

的非负指数

i,j,k

确定

A term is determined by nonnegative exponents

i,j,k

whose sum is

10

用隔板法计算

i+j+k=10

的解数

Count the solutions of

i+j+k=10

by stars and bars

每个不同项都是

a^ib^jc^k

，其中非负整数满足

i+j+k=10\text{。}

由隔板法，这样的三元组个数为

\binom{10+3-1}{3-1}=\binom{12}{2}=66\text{。}

所以正确答案为 D。

Each distinct term is

a^ib^jc^k

for nonnegative integers satisfying

i+j+k=10.

By stars and bars, the number of such triples is

\binom{10+3-1}{3-1}=\binom{12}{2}=66.

Thus, the correct answer is D.

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