In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If 14 boys and 10 girls are competing, how many different ways could the six medals possibly be given out? Answer Attempt 1 out of 2

To find the number of different ways to award the medals, we first calculate the combinations for boys and girls separately and then multiply the results. For the boys, we need to select 3 winners from 14 competitors. The number of ways to select and arrange the top 3 can be calculated using permutations: - For gold, there are 14 choices. - For silver, there are 13 remaining choices after gold is awarded. - For bronze, there are 12 remaining choices after silver is awarded. So, for boys, it's \( 14 \times 13 \times 12 \). For the girls, the calculation is similar: - For gold, there are 10 choices. - For silver, 9 choices remaining. - For bronze, 8 choices remaining. So, for girls, it's \( 10 \times 9 \times 8 \). Now, we multiply the two results: - Total combinations for boys = \( 14 \times 13 \times 12 = 2184 \). - Total combinations for girls = \( 10 \times 9 \times 8 = 720 \). Therefore, the total combinations for assigning medals to both boys and girls becomes \( 2184 \times 720 = 1,572,768 \). So, the total ways the six medals could be awarded is 1,572,768.