Question

arturo is designing a new board game, and is trying to figure out all the possible outcomes. how many different possible outcomes are there if he spins a spinner with 5 equal - sized sections labeled monday, tuesday, wednesday, thursday, friday, spins a spinner with four equal - sized sections labeled red, green, blue, orange, and rolls a fair die in the shape of a cube that has six sides labeled 1 to 6?

Explanation:

Step1: Identify the number of outcomes for each event

• Spinner 1 (days): It has 5 equal - sized sections, so the number of outcomes for spinning this spinner, let's call it \( n_1 = 5 \).
• Spinner 2 (colors): It has 4 equal - sized sections, so the number of outcomes for spinning this spinner, \( n_2=4 \).
• Die roll: A fair die has 6 sides, so the number of outcomes for rolling the die, \( n_3 = 6 \).

Step2: Use the multiplication principle of counting

The multiplication principle states that if there are \( m_1\) ways to do the first task, \( m_2\) ways to do the second task, and \( m_3\) ways to do the third task, then the total number of ways to do all three tasks together is \( m_1\times m_2\times m_3 \).

So the total number of possible outcomes \( N=n_1\times n_2\times n_3\)

Substitute \( n_1 = 5 \), \( n_2 = 4 \) and \( n_3=6 \) into the formula:

\( N=5\times4\times6 \)

First, calculate \( 5\times4 = 20 \)

Then, calculate \( 20\times6=120 \)

Answer:

The total number of different possible outcomes is 120.