There are 12 boys and 15 girls. How many different dancing groups can be…

Concept: when two selections are made independently from two different groups, the number of ways to choose r items out of n items uses the combination formula nCr = n! / (r! x (n - r)!). By the fundamental counting principle, when two independent selections are combined, the total number of ways to make both selections together equals the PRODUCT of the two individual combination counts.

1. Number of ways to choose 2 boys out of 12: 12C2 = 12! / (2! x 10!) = (12 x 11) / (2 x 1) = 66.

2. Number of ways to choose 3 girls out of 15: 15C3 = 15! / (3! x 12!) = (15 x 14 x 13) / (3 x 2 x 1) = 455.

3. Since the boy-selection and the girl-selection are independent, multiply the two counts: 66 x 455 = 30030.

Cross-check: recomputing 12C2 directly as (12 x 11) / 2 = 66 and 15C3 as (15 x 14 x 13) / 6 = 455 reproduces the same intermediate values; their product 66 x 455 gives 30030, matching the option value exactly.

Hence, the total number of different dancing groups that can be formed is 30030.