Final answer:
To determine the number of study groups of 3 girls and 2 boys that can be formed from 32 girls and 18 boys, combinations are calculated separately for girls (4,960) and boys (153), and then multiplied to give a total of 758,880 possible study groups.
Step-by-step explanation:
The question involves calculating the number of possible study groups that can be formed from a class of 32 girls and 18 boys, where each study group consists of 3 girls and 2 boys. This problem falls under the subject of combinatorics, a branch of mathematics dealing with the counting, combination, and permutation of sets of elements.
To solve this problem, we need to use the combination formula to find out how many different ways we can choose 3 girls from the 32 available, and 2 boys from the 18 available, and then multiply those two results to get the total number of possible study groups. The combination formula for choosing k elements from a set of n elements is given as C(n, k) = n! / (k! * (n - k)!), where "!" represents factorial.
The number of ways to choose 3 girls from 32 is C(32, 3) = 32! / (3! * (32 - 3)!) and for the boys, C(18, 2) = 18! / (2! * (18 - 2)!). Calculating these gives us the number of possible groups for girls and boys separately:
- C(32, 3) = 4,960 combinations of girls
- C(18, 2) = 153 combinations of boys
To find the total number of possible study groups, we multiply these two results together:
Total study groups = 4,960 * 153 = 758,880
Thus, there can be 758,880 different study groups each consisting of 3 girls and 2 boys formed from this class.