5. Stars & Bars: problem set · Mastering AMC 10/12

Problem 5.5.1

AMC

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

Problem 5.5.2

AMC 10

When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as

\frac{n}{6^7},

where $n$ is a positive integer. What is $n$ ?

Problem 5.5.3

AMC 10

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

Problem 5.5.4

AMC 10

A seven-digit positive integer is chosen at random. What is the probability that the number is divisible by 11, given that the sum of its digits is 61?

Problem 5.5.5

AMC 10Challenge

This problem also appears in the book as 3.3.18.

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product

n = f_1 \cdot f_2 \cdots f_k,

where $k \ge 1$ , the $f_i$ are integers strictly greater than 1, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as $6$ , $2 \cdot 3$ , and $3 \cdot 2$ , so $D(6) = 3$ . What is $D(96)$ ?

Problem 5.5.6(additional)

Omega Learn

Find the number of solutions to $a + b + c + d < 19$ where $a, b, c, d$ are positive integers, $a$ and $c$ must be odd, and $b$ and $d$ must be even.

Problem 5.5.7(additional)

How many terms will be in the expansion of $(w + x + y + z)^{10}$ after simplifying?

Problem 5.5.8(additional)

AMC 12

The expression

(x + y + z)^{2006} + (x - y - z)^{2006}

is simplified by expanding it and combining like terms. How many terms are in the simplified expression?

Problem 5.5.9(additional)

AIME

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 5.5.10(additional)

AIME

A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m + n$ .

Problem 5.5.11(additional)

In how many ways can 8 students and 3 teachers sit in the front row if, between every pair of teachers, there must be at least 2 students?

Problem 5.5.12(additional)

AIME

Consider the paths of length 16 that go from the lower left corner to the upper right corner of an $8 \times 8$ grid. Find the number of such paths that change direction exactly 4 times.

Stars & Bars: all problems