Question

in a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. if 15 boys and 11 girls are competing, how many different ways could the six medals possibly be given out?

Explanation:

Step1: Calculate permutations for boys

We need to find the number of ways to give 3 medals (gold, silver, bronze) to 15 boys. This is a permutation problem since the order of medals matters. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 15 \) and \( r=3 \).
\( P(15, 3)=\frac{15!}{(15 - 3)!}=\frac{15!}{12!}=15\times14\times13 = 2730 \)

Step2: Calculate permutations for girls

We need to find the number of ways to give 3 medals to 11 girls. Using the permutation formula with \( n = 11 \) and \( r = 3 \).
\( P(11, 3)=\frac{11!}{(11 - 3)!}=\frac{11!}{8!}=11\times10\times9=990 \)

Step3: Calculate total number of ways

To find the total number of ways to give out all six medals, we multiply the number of ways for boys and the number of ways for girls (by the multiplication principle of counting).
Total ways \(=P(15, 3)\times P(11, 3)=2730\times990 = 2702700 \)

Answer:

2702700