Question

in a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. if 8 boys and 7 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 permutations of 8 boys taken 3 at a time. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 8 \) and \( r=3 \).
\( P(8, 3)=\frac{8!}{(8 - 3)!}=\frac{8!}{5!}=\frac{8\times7\times6\times5!}{5!}=8\times7\times6 = 336 \)

Step2: Calculate permutations for girls' medals

We need to find the number of permutations of 7 girls taken 3 at a time. Using the permutation formula with \( n = 7 \) and \( r = 3 \).
\( P(7, 3)=\frac{7!}{(7 - 3)!}=\frac{7!}{4!}=\frac{7\times6\times5\times4!}{4!}=7\times6\times5=210 \)

Step3: Find total number of ways to give out medals

To find the total number of ways to give out all six medals, we multiply the number of ways to give out boys' medals and girls' medals.
Total ways \(=P(8, 3)\times P(7, 3)=336\times210 = 70560 \)

Answer:

\( 70560 \)