Question

in a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. if 9 boys and 8 girls are competing, how many different ways could the six medals possibly be given out?

Explanation:

Step1: Calculate permutations for boys' medals

We need to find the number of ways to award 3 medals (gold, silver, bronze) to 9 boys. This is a permutation problem, calculated using the permutation formula \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 9 \) and \( r=3 \).
\( P(9, 3)=\frac{9!}{(9 - 3)!}=\frac{9!}{6!}=9\times8\times7 = 504 \)

Step2: Calculate permutations for girls' medals

We need to find the number of ways to award 3 medals to 8 girls. Using the permutation formula with \( n = 8 \) and \( r = 3 \).
\( P(8, 3)=\frac{8!}{(8 - 3)!}=\frac{8!}{5!}=8\times7\times6=336 \)

Step3: Calculate total number of ways

To find the total number of ways to award all six medals, we multiply the number of ways to award boys' medals and girls' medals.
Total ways \(= P(9, 3)\times P(8, 3)=504\times336 = 169344 \)

Answer:

169344