Final answer:
The number of ways a coach can choose a team of 10 players from a roster with the requirement of including at least 3 girls is calculated using combinations for each scenario of girls and boys that meet the team size criteria, then summing those results.
Step-by-step explanation:
To determine how many ways a coach can choose a team of 10 players from a roster of 12 boys and 13 girls, with the requirement that at least 3 girls must be chosen, we use combinations to consider the different possible scenarios for team composition: choosing 3, 4, 5, 6, or 7 girls (since choosing more than 7 girls would result in less than 3 boys, which isn't allowed as you need 10 players in total).
- Choosing 3 girls out of 13 can be done in C(13,3) ways.
- Choosing 7 boys out of 12 can be done in C(12,7) ways.
- Similarly, we calculate the combinations for choosing 4, 5, 6, or 7 girls and the corresponding number of boys.
- The total number of ways the team can be chosen is the sum of the products of these combinations for each scenario.
For example, the number of ways to choose a team with 3 girls and 7 boys is C(13,3) * C(12,7), and we would calculate this for all combinations that meet the criteria and sum them up for the final answer.