Final answer:
To find the number of different options for winners in a talent contest with 6 boys and 6 girls, calculate the permutations for each and then multiply them. The answer is 120 ways for the boys, multiplied by 120 ways for the girls, equaling 14400 different possible outcomes.
Step-by-step explanation:
The question involves determining the number of different possible outcomes for choosing the top 3 winners among the boys and the top 3 winners among the girls in a talent contest. To do this, we use combinatorics, specifically the concept of permutations, because the order of the winners matters.
For the boys, there are 6 contestants and 3 winners to be chosen in order. The number of ways this can be done is a permutation of 6 items taken 3 at a time (denoted as P(6,3)). The same calculation applies to the girls.
Therefore, the total number of options for winners is P(6,3) for the boys multiplied by P(6,3) for the girls. Calculating, we get:
- P(6,3) = 6 × 5 × 4 = 120 ways for the boys,
- P(6,3) = 6 × 5 × 4 = 120 ways for the girls.
So, the total number of different options for winners is 120 × 120 = 14400.