To determine how many different toys there can be, we use the concept of the multiplication rule in combinatorics. The multiplication rule states that if there are [tex]\( m \)[/tex] ways to do one thing and [tex]\( n \)[/tex] ways to do another, then there are [tex]\( m \times n \)[/tex] ways to do both.
Let's go through this step-by-step:
1. Identify the number of ways to do the first task (choosing a color):
- There are [tex]\( m \)[/tex] different colors.
2. Identify the number of ways to do the second task (choosing a size):
- There are [tex]\( n \)[/tex] different sizes.
3. Determine the total number of different combinations:
- For each of the [tex]\( m \)[/tex] colors, there are [tex]\( n \)[/tex] possible sizes. Hence, to find the total number of different toys, we multiply the number of ways to choose a color by the number of ways to choose a size.
Therefore, the total number of different toys is calculated by:
[tex]\[ m \times n \][/tex]
For example, if the toy comes in [tex]\( 5 \)[/tex] colors and [tex]\( 4 \)[/tex] sizes, we can calculate the number of different toys as follows:
[tex]\[ 5 \times 4 = 20 \][/tex]
Thus, there are [tex]\( 20 \)[/tex] different toys possible.
Among the given choices:
- [tex]\( m^n \)[/tex] suggests an exponential relationship which does not apply here.
- [tex]\( m \times n \)[/tex] correctly represents the multiplication rule.
- [tex]\( m + n \)[/tex] suggests adding the ways, which is not correct for determining combinations.
- [tex]\( m(n-1) \)[/tex] is incorrect as it suggests subtracting one size from the multiplication.
The correct answer for how many ways there are to do both tasks is:
[tex]\[ m \times n \][/tex]
