Number of ways to form a 5 member team from 7 boys and 6 girls such that the team includes at least 3 girls

Introduction / Context:
This selection problem uses combinations with an additional constraint on the composition of the team. The team must contain at least three girls, which means we must consider multiple cases for how many girls are included and then sum those cases.

Given Data / Assumptions:

• Total boys available: 7.
• Total girls available: 6.
• Team size: 5 members.
• The team must have at least 3 girls.
• All boys and girls are distinct individuals.

Concept / Approach:
At least three girls means the team can have exactly 3 girls, exactly 4 girls, or exactly 5 girls. For each case, we count the number of ways to select the required number of girls and boys using combinations, then add the results because the cases are mutually exclusive.

Step-by-Step Solution:
Step 1: Case 1 is exactly 3 girls and 2 boys.Step 2: Select 3 girls from 6 in 6C3 ways and 2 boys from 7 in 7C2 ways.Step 3: Compute 6C3 = 20 and 7C2 = 21, so Case 1 count = 20 * 21 = 420.Step 4: Case 2 is exactly 4 girls and 1 boy.Step 5: Select 4 girls from 6 in 6C4 ways and 1 boy from 7 in 7C1 ways.Step 6: Compute 6C4 = 15 and 7C1 = 7, so Case 2 count = 15 * 7 = 105.Step 7: Case 3 is exactly 5 girls and 0 boys.Step 8: Select 5 girls from 6 in 6C5 ways and 0 boys from 7 in 7C0 ways.Step 9: Compute 6C5 = 6 and 7C0 = 1, so Case 3 count = 6 * 1 = 6.Step 10: Total valid teams = 420 + 105 + 6 = 531.

Verification / Alternative check:
You can verify by ensuring that none of the cases exceed the available boys or girls and that the total team size is always 5. All three cases satisfy these constraints, and their counts add directly because they do not overlap.

Why Other Options Are Wrong:

• 427 and 651 are close in size but do not match the precise sum of 420, 105, and 6.
• 714 is too large and would require counting extra invalid cases where there are fewer than 3 girls.

Common Pitfalls:
Students sometimes include teams with 2 girls by mistake, misreading at least three girls as at most three girls. Another common error is to forget the case with 5 girls and 0 boys. Always write out all valid compositions and check that their sum matches the team size.

Final Answer:
The number of ways to form such a team is 531, so the correct answer is 531.