Question

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:

$11$

$33$

$55$

$66$

$132$

Answer: D

Concepts:stars and bars combinations polynomial

Difficulty rating: 1340

Small Hint:

A term is determined by nonnegative exponents $i,j,k$ whose sum is $10$

Big Hint:

Count the solutions of $i+j+k=10$ by stars and bars

Solution:

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.

1958 AMC 12 Problem 49

Answer: D

Difficulty rating: 1340

Small Hint:

Big Hint:

Solution:

Problem 49 in Other Years