In how many ways can 7 people be divided into 3 groups if one group must have 3

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Question: In how many ways can 7 people be divided into 3 groups if one group must have 3 people?

Options:

Correct Answer: 300

Solution:

Choose 3 people from 7: C(7, 3) = 35. The remaining 4 can be divided into 2 groups in 2 ways. Total = 35 * 2 = 70.

In how many ways can 7 people be divided into 3 groups if one group must have 3 people?

In how many ways can 7 people be divided into 3 groups if one group must have 3 people?

Step 1: We have 7 people and we need to choose 3 people to form the first group.
Step 2: To find out how many ways we can choose 3 people from 7, we use the combination formula C(7, 3). This is calculated as 7! / (3! * (7-3)!) which equals 35.
Step 3: After choosing 3 people for the first group, we have 4 people left.
Step 4: We need to divide these 4 remaining people into 2 groups. There are 2 ways to do this: either we can have 2 people in one group and 2 in the other, or we can have 1 person in one group and 3 in the other.
Step 5: Since we are only interested in the case where the groups are of equal size (2 and 2), we can simply divide the 4 people into 2 groups in 2 ways.
Step 6: Now, we multiply the number of ways to choose the first group (35) by the number of ways to divide the remaining people (2).
Step 7: Therefore, the total number of ways to divide the 7 people into 3 groups, with one group having 3 people, is 35 * 2 = 70.

Combinatorics – The problem involves choosing a subset of people from a larger group and then dividing the remaining individuals into groups.
Group Division – Understanding how to partition a set of individuals into specified group sizes.