Out of 3 girls and 6 boys, a group of three members is to be formed so that at least one member is a girl. In how many different ways can such a group be formed?

Introduction / Context:
This is a combinatorics question about selecting a small group from a larger set. We have boys and girls, and we must form a three member group with the condition that there is at least one girl in the group.

Given Data / Assumptions:

• Girls available = 3.
• Boys available = 6.
• Total people = 3 + 6 = 9.
• Group size = 3 members.
• The group must contain at least one girl, so all boy groups are not allowed.

Concept / Approach:
It is often easier to count all possible groups and then subtract the groups that violate the condition. Here, we can count all 3 member groups from 9 people and then subtract the groups with no girl, which are formed entirely from boys. The remaining groups automatically have at least one girl.

Step-by-Step Solution:
Total ways to choose any 3 people from 9 = 9C3.Compute 9C3 = 9 * 8 * 7 / (3 * 2 * 1) = 84.Now count the disallowed groups, which contain no girl. These consist of 3 boys chosen from 6.Number of all boy groups = 6C3.Compute 6C3 = 6 * 5 * 4 / (3 * 2 * 1) = 20.Valid groups with at least one girl = total groups - all boy groups = 84 - 20 = 64.

Verification / Alternative check:
We can verify by directly counting by cases based on the number of girls in the group: exactly one girl, exactly two girls, exactly three girls. Exactly one girl: 3C1 * 6C2 = 3 * 15 = 45. Exactly two girls: 3C2 * 6C1 = 3 * 6 = 18. Exactly three girls: 3C3 * 6C0 = 1 * 1 = 1. Sum = 45 + 18 + 1 = 64, which agrees with our earlier calculation.

Why Other Options Are Wrong:
The value 84 counts all possible groups without applying the condition, so it includes the 20 all boy groups. The number 20 counts only the all boy groups and therefore is not what we need. The value 56 does not match any consistent breakdown of the valid cases and comes from an incorrect subtraction or counting method. Only 64 satisfies the constraint correctly.

Common Pitfalls:
A frequent mistake is to count only groups with exactly one girl and forget the cases of two or three girls. Another error is to subtract the wrong number of all boy groups because of errors in calculating 6C3. Writing out the combination formula and computing carefully, or using the complement method as described, helps prevent such miscounts.

Final Answer:
The number of different groups that can be formed with at least one girl is 64.