In how many ways can 4 people be chosen and arranged in a straight line, if there are 5 people from whom to choose?
A shirt company has 3 designs that can be made with short or long sleeves. There are 5 color patterns available. How many different types of shirts are available from this company?
5P4 = 120
3*2*5
To find the number of ways to choose and arrange 4 people in a straight line from 5 people total, we can use the concept of permutations.
The number of ways to choose and arrange 4 people out of 5 is given by the formula for permutations: nPr = n! / (n - r)!
In this case, n (total number of people) is 5 and r (number of people chosen) is 4.
So, the number of ways to choose and arrange 4 people from 5 is:
5P4 = 5! / (5 - 4)!
= 5! / 1!
= 5 * 4 * 3 * 2 * 1 / 1
= 5 * 4 * 3 * 2
= 120
Therefore, there are 120 ways to choose and arrange 4 people in a straight line from a group of 5 people.
Now, let's move on to the second question.
To calculate the number of different types of shirts available from the company, we need to multiply the number of design options by the number of color patterns.
Given:
- 3 designs (short or long sleeves)
- 5 color patterns
The number of different types of shirts available is obtained by multiplying these two numbers together:
Number of different types of shirts = Number of designs * Number of color patterns
Number of different types of shirts = 3 * 5
= 15
Therefore, from this company, there are 15 different types of shirts available.
To find the number of ways 4 people can be chosen and arranged in a straight line from a group of 5 people, we can use the concept of permutations.
The number of ways to choose and arrange the 4 people is given by the formula for permutations:
P(n, r) = n! / (n - r)!
In this case, n = 5 (the total number of people to choose from) and r = 4 (the number of people we want to choose and arrange).
Plugging the values into the formula:
P(5, 4) = 5! / (5 - 4)!
= 5! / 1!
= 5 * 4 * 3 * 2
= 120
Therefore, there are 120 ways to choose and arrange 4 people in a straight line from a group of 5 people.
Now, to find the number of different types of shirts available from the company, we need to multiply the number of design options by the number of color patterns.
In this case, there are 3 designs and 5 color patterns. To find the total number of different types of shirts, we multiply these numbers:
Total = Number of designs × Number of color patterns
= 3 × 5
= 15
Therefore, there are 15 different types of shirts available from this company.