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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 6 boys and 15 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 $6$ boys and $15$ girls are competing, how many different ways could the six medals possibly be given out? $\newline$ Answer:

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Counting principle

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

6

boys and

15

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

\newline

Answer:

Calculate Boys Medals: Calculate the number of ways to give out medals to the boys. $\newline$ Since there are $6$ boys and $3$ medals (gold, silver, bronze), we use permutations because the order matters (gold is different from silver, which is different from bronze). $\newline$ Number of ways to give out medals to the boys = $6P3 = \frac{6!}{(6-3)!}$
Calculate Boys Medals Result: Perform the calculation for the number of ways to give out medals to the boys. $\newline$ $6! / 3! = (6 \times 5 \times 4 \times 3 \times 2 \times 1) / (3 \times 2 \times 1) = 6 \times 5 \times 4 = 120$
Calculate Girls Medals: Calculate the number of ways to give out medals to the girls. $\newline$ Since there are $15$ girls and $3$ medals, we again use permutations. $\newline$ Number of ways to give out medals to the girls = $15P3 = \frac{15!}{(15-3)!}$
Calculate Girls Medals Result: Perform the calculation for the number of ways to give out medals to the girls. $15! / 12! = (15 \times 14 \times 13 \times 12 \times 11 \times \ldots \times 1) / (12 \times 11 \times \ldots \times 1) = 15 \times 14 \times 13 = 2730$
Calculate Total Ways: Calculate the total number of different ways the six medals can be given out. $\newline$ Since the medal distribution for boys and girls are independent events, we multiply the number of ways for boys by the number of ways for girls. $\newline$ Total number of ways $=$ Number of ways for boys $\times$ Number of ways for girls
Calculate Total Ways Result: Perform the final calculation to find the total number of different ways the six medals can be given out. $\newline$ Total number of ways = $120$ (for boys) $\times$ $2730$ (for girls) = $327600$

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