In how many ways can 8 people be divided into 2 groups of 3 and 5 people respectively?
Thanks again to anyone who helps. Answer is 56
8C3 = 8C5 = 56
To find the number of ways to divide 8 people into two groups of 3 and 5 people respectively, we can use combination formula.
Step 1: Selecting the group of 3 people:
To choose 3 people out of 8, we can use the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Substituting the values, we get:
C(8, 3) = 8! / (3! * (8-3)!)
= 8! / (3! * 5!)
= 8 * 7 * 6 / (3 * 2 * 1)
= 56
So, there are 56 ways to select the group of 3 people.
Step 2: Once the group of 3 people is selected, the remaining 5 people will automatically form the other group.
Therefore, there is only 1 way to select the group of 5 people.
Step 3: Since the group of 3 people and the group of 5 people are selected independently, we can multiply the number of ways from step 1 and step 2.
56 * 1 = 56
Hence, there are 56 ways to divide 8 people into 2 groups of 3 and 5 people respectively.
To solve this problem, we can use combinations.
To find the number of ways to divide 8 people into 2 groups of 3 and 5, we can first choose 3 people from the 8 to form the first group, and the remaining 5 will form the second group.
The number of ways to choose 3 people from 8 is denoted as "8 choose 3" and can be calculated using the formula:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items to choose from, and r is the number of items to choose.
So, for our case, we have:
C(8, 3) = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Hence, there are 56 ways to divide 8 people into 2 groups of 3 and 5 people respectively.