A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected, if the team has

• No girl.
• At least one boy and one girl.
• At least three girls.

To tackle the problem of selecting a team of 5 members from a group of 4 girls and 7 boys under different conditions, we can break it down into three distinct scenarios. Each scenario will require a different approach to counting the combinations. Let’s go through each case step by step.

Scenario 1: Selecting a Team with No Girls

In this scenario, we need to select a team of 5 members consisting solely of boys. Since there are 7 boys available, we can use the combination formula to determine how many ways we can choose 5 boys from these 7.

The combination formula is given by:

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.

For our case:

• n = 7 (the number of boys)
• r = 5 (the number of boys we want to select)

Calculating this gives us:

C(7, 5) = 7! / (5! * (7 - 5)!) = 7! / (5! * 2!) = (7 × 6) / (2 × 1) = 21

Scenario 2: Selecting a Team with At Least One Boy and One Girl

Here, we need to ensure that our team includes at least one boy and one girl. To find the total combinations, we can calculate the total ways to select any 5 members from the group and then subtract the cases that do not meet the criteria.

First, calculate the total ways to select 5 members from 11 (4 girls + 7 boys):

C(11, 5) = 11! / (5! * (11 - 5)!) = 11! / (5! * 6!) = (11 × 10 × 9 × 8) / (4 × 3 × 2 × 1) = 462

Next, we need to subtract the cases where there are no girls (which we calculated earlier as 21) and the cases where there are only girls. Since there are only 4 girls, it is impossible to select a team of 5 girls. Therefore, we only subtract the 21 ways:

Total valid selections = 462 - 21 = 441

Scenario 3: Selecting a Team with At Least Three Girls

For this scenario, we can have either 3, 4, or 5 girls in the team. We will calculate the combinations for each case separately and then sum them up.

Case 1: 3 Girls and 2 Boys

We can choose 3 girls from 4 and 2 boys from 7:

C(4, 3) = 4

C(7, 2) = 21

Thus, the total for this case is:

4 × 21 = 84

Case 2: 4 Girls and 1 Boy

We can choose all 4 girls and 1 boy from 7:

C(4, 4) = 1

C(7, 1) = 7

Thus, the total for this case is:

1 × 7 = 7

Final Calculation

Now, we sum the totals from both cases:

Total for at least 3 girls = 84 + 7 = 91

Summary of Results

• Ways to select a team with no girls: 21
• Ways to select a team with at least one boy and one girl: 441
• Ways to select a team with at least three girls: 91

By breaking down the problem into manageable parts, we can clearly see how to approach each scenario and arrive at the correct answers. If you have any further questions or need clarification on any of the steps, feel free to ask!