How Many Ways Can a Basketball Team of 5 Players Be Selected from 6 Girls and 7 Boys?

The question of selecting a basketball team of 5 players from 6 girls and 7 boys, while adhering to specific constraints, is an interesting problem that involves the use of combinatorics. Let's break down the problem and explore the methods to solve it.

Problem Statement

A basketball team of 5 players is to be picked from 6 girls and 7 boys. However, there are two constraints: the team must have more girls than boys and it must include at least one boy. The objective is to determine the number of different teams that can be formed under these conditions.

Solution

To solve this problem, we need to consider all possible combinations that meet the given criteria and then sum them up. The constraints provided imply that the number of girls in the team must be 4 or 5, and there must be 1 or 2 boys. This is because having 3 or more boys would violate the requirement for more girls than boys.

Step 1: Selecting 4 Girls and 1 Boy

The first scenario involves selecting 4 girls from 6 and 1 boy from 7.

binom{6}{4} times binom{7}{1} frac{6!}{4!2!} times 7 15 times 7 105

Step 2: Selecting 3 Girls and 2 Boys

The second scenario involves selecting 3 girls from 6 and 2 boys from 7.

binom{6}{3} times binom{7}{2} frac{6!}{3!3!} times frac{7!}{2!5!} 20 times 21 420

Step 3: Selecting 2 Girls and 3 Boys

The third scenario involves selecting 2 girls from 6 and 3 boys from 7.

binom{6}{2} times binom{7}{3} frac{6!}{2!4!} times frac{7!}{3!4!} 15 times 35 525

Step 4: Selecting 1 Girl and 4 Boys

The fourth scenario involves selecting 1 girl from 6 and 4 boys from 7.

binom{6}{1} times binom{7}{4} 6 times frac{7!}{4!3!} 6 times 35 210

Summarizing the Results

The total number of ways to form a basketball team under the given constraints is the sum of all the above scenarios.

Total ways 105 420 525 210 1260

Alternative Method Using Combinatorics

To validate our solution, we can use the total number of ways to form a team of 5 players from 13 candidates (6 girls 7 boys) and subtract the invalid teams (i.e., teams with no boys or no girls).

Total ways to form a team of 5 from 13 candidates:

binom{13}{5} frac{13!}{5!(13-5)!} 1287

Teams with no boys (all girls):

binom{6}{5} 6

Teams with no girls (all boys):

binom{7}{5} 21

Valid teams Total ways - Teams with no boys - Teams with no girls

Valid teams 1287 - 6 - 21 1260

Conclusion

Thus, the number of different ways to select a basketball team of 5 players from 6 girls and 7 boys, ensuring that there are more girls than boys and at least one boy, is 1260.