Question

1.1 to 1.3 mastery check
indicate whether the prompt requires permutations, combinations, or the fundamental counting principle and explain. then, solve the problem.

1. ryan and prancer are planning trips to three countries this year. there are 7 countries they would like to visit. how many ways can they plan the three countries to visit?

type: ____ explanation: ____

1. five rooms in a house need to be painted. each room can be painted either green, yellow, blue, or white. in how many different ways can the rooms be painted?

type: ____ explanation: ____

1. the student body of 790 people is going to choose a president and a vice president. how many ways can the roles be filled?

type: ____ explanation: ____

1. there are 12 players on a softball team. the batting order can only include nine players. in how many ways can you choose a batting order?

type: ____ explanation: ____

Explanation:

Problem 1:

Step1: Identify the type

This is a combination problem because the order of visiting the countries does not matter (we are just choosing 3 countries out of 7 to visit, and the sequence of visiting them isn't specified to be important here). The formula for combinations is \( C(n, r)=\frac{n!}{r!(n - r)!} \), where \( n = 7 \) (total countries) and \( r=3 \) (countries to visit).

Step2: Calculate the combination

Substitute \( n = 7 \) and \( r = 3 \) into the combination formula:
\[

$$\begin{align*} C(7,3)&=\frac{7!}{3!(7 - 3)!}\\ &=\frac{7!}{3!4!}\\ &=\frac{7\times6\times5\times4!}{3\times2\times1\times4!}\\ &=\frac{7\times6\times5}{3\times2\times1}\\ &=\frac{210}{6}\\ & = 35 \end{align*}$$

\]

Step1: Identify the type

This is a problem using the Fundamental Counting Principle. Each room has 4 choices (green, yellow, blue, white), and there are 5 rooms. The Fundamental Counting Principle states that if there are \( m_1\) ways to do the first task, \( m_2\) ways to do the second task, \(\cdots\), \( m_k\) ways to do the \( k\)-th task, then the total number of ways to do all \( k\) tasks is \( m_1\times m_2\times\cdots\times m_k \). Here, for each of the 5 rooms (tasks), there are 4 choices (ways), so we multiply the number of choices for each room.

Step2: Calculate the total number of ways

Using the Fundamental Counting Principle, the total number of ways to paint 5 rooms with 4 color choices per room is \( 4\times4\times4\times4\times4=4^5 \).
\[
4^5=1024
\]

Step1: Identify the type

This is a permutation problem because the order matters (choosing a president and then a vice - president: the person chosen as president is different from the person chosen as vice - president, so the order of selection matters). The formula for permutations is \( P(n,r)=\frac{n!}{(n - r)!} \), where \( n = 750 \) (total number of students) and \( r = 2 \) (positions: president and vice - president).

Step2: Calculate the permutation

Substitute \( n = 750 \) and \( r = 2 \) into the permutation formula:
\[

$$\begin{align*} P(750,2)&=\frac{750!}{(750 - 2)!}\\ &=\frac{750!}{748!}\\ &=\frac{750\times749\times748!}{748!}\\ &=750\times749\\ &=561750 \end{align*}$$

\]

Answer:

The number of ways is 35. The type is combinations, and the explanation is that the order of visiting the countries does not matter, so we use the combination formula \( C(n,r)=\frac{n!}{r!(n - r)!} \) with \( n = 7 \) and \( r = 3 \) to get 35.

Problem 2: