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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 7 boys and 14 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 $7$ boys and $14$ girls are competing, how many different ways could the six medals possibly be given out? $\newline$ Answer:

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Experimental probability

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

7

boys and

14

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

\newline

Answer:

Calculate Boys' Medal Distribution: Step $1$ : Determine the number of ways to distribute the medals among the boys. $\newline$ Since there are $7$ boys competing for $3$ medals (gold, silver, and bronze), we use permutations because the order matters. The number of ways to arrange $3$ medals among $7$ boys is given by the permutation formula $P(n, k) = \frac{n!}{(n - k)!}$ , where $n$ is the total number of boys and $k$ is the number of medals. $\newline$ Calculation: P(\(7\), \(3\)) = \frac{\(7\)!}{(\(7\) - \(3\))!} = \frac{\(7\)!}{\(4\)!} = \frac{(\(7\) \times \(6\) \times \(5\) \times \(4\)!)}{\(4\)!} = \(7\) \times \(6\) \times \(5\) = \(210\)
Calculate Girls' Medal Distribution: Step \(2\): Determine the number of ways to distribute the medals among the girls.\(\newline\)Since there are \(14\) girls competing for \(3\) medals, we use the same permutation formula as in Step \(1\). The number of ways to arrange \(3\) medals among \(14\) girls is \(P(14, 3)\).\(\newline\)Calculation: P( $14$ , $3$ ) = \frac{ $14$ !}{( $14$ - $3$ )!} = \frac{ $14$ !}{ $11$ !} = \frac{( $14$ \times $13$ \times $12$ \times $11$ !)}{ $11$ !} = $14$ \times $13$ \times $12$ = $2184$
Calculate Total Medal Distribution: Step $3$ : Calculate the total number of ways to distribute all six medals. $\newline$ To find the total number of ways to distribute all six medals, we multiply the number of ways to distribute the medals among the boys by the number of ways to distribute the medals among the girls. This is because each arrangement of medals for the boys can be paired with each arrangement of medals for the girls. $\newline$ Calculation: Total ways = Ways for boys $\times$ Ways for girls = $210 \times 2184 = 458640$

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