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In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If 13 boys and 5 girls are competing, how many different ways could the six medals possibly be given out?
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In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If $13$ boys and $5$ girls are competing, how many different ways could the six medals possibly be given out? $\newline$ Answer:

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Compound events: find the number of outcomes

Full solution

Q. In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If

13

boys and

5

girls are competing, how many different ways could the six medals possibly be given out?

\newline

Answer:

Calculate Boys' Medal Permutations: First, we need to calculate the number of ways the medals can be given out to the boys. Since there are $13$ boys and $3$ medals (gold, silver, bronze), we use permutations because the order in which the medals are awarded matters. $\newline$ The number of ways to award $3$ medals to $13$ boys is given by the permutation formula: $\newline$ $P(n, k) = \frac{n!}{(n - k)!}$ $\newline$ where $n$ is the total number of boys and $k$ is the number of medals.
Calculate Girls' Medal Permutations: For the boys, we calculate the permutation: $\newline$ $P(13, 3) = \frac{13!}{(13 - 3)!}$ $\newline$ $P(13, 3) = \frac{13!}{10!}$ $\newline$ $P(13, 3) = 13 \times 12 \times 11$ $\newline$ $P(13, 3) = 1716$ $\newline$ So, there are $1716$ different ways to award the medals to the boys.
Calculate Total Ways: Next, we calculate the number of ways the medals can be given out to the girls. Since there are $5$ girls and $3$ medals, we again use permutations. $\newline$ $P(5, 3) = \frac{5!}{(5 - 3)!}$ $\newline$ $P(5, 3) = \frac{5!}{2!}$ $\newline$ $P(5, 3) = 5 \times 4 \times 3$ $\newline$ $P(5, 3) = 60$ $\newline$ So, there are $60$ different ways to award the medals to the girls.
Perform Multiplication: Finally, to find the total number of different ways the six medals can be given out, we multiply the number of ways for the boys by the number of ways for the girls. $\newline$ Total ways $= \text{Ways for boys} \times \text{Ways for girls}$ $\newline$ Total ways $= 1716 \times 60$
Perform Multiplication: Finally, to find the total number of different ways the six medals can be given out, we multiply the number of ways for the boys by the number of ways for the girls. $\newline$ Total ways $= \text{Ways for boys} \times \text{Ways for girls}$ $\newline$ Total ways $= 1716 \times 60$ We perform the multiplication to find the total number of ways: $\newline$ Total ways $= 1716 \times 60$ $\newline$ Total ways $= 102960$ $\newline$ So, there are $102960$ different ways the six medals can be given out.

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