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

In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If 1212 boys and 66 girls are competing, how many different ways could the six medals possibly be given out?\newlineAnswer:

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Q. In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If 1212 boys and 66 girls are competing, how many different ways could the six medals possibly be given out?\newlineAnswer:
  1. Determine boys' medal permutations: Determine the number of ways to award medals to the boys.\newlineSince there are 1212 boys competing and 33 medals to be awarded (gold, silver, and bronze), we need to calculate the number of permutations of 1212 boys taken 33 at a time.\newlineThe formula for permutations is P(n,k)=n!(n−k)!P(n, k) = \frac{n!}{(n - k)!} where nn is the total number of items and kk is the number of items to choose.\newlineFor the boys, n=12n = 12 and k=3k = 3.\newlineP(12,3)=12!(12−3)!=12!9!=12×11×10P(12, 3) = \frac{12!}{(12 - 3)!} = \frac{12!}{9!} = 12 \times 11 \times 10
  2. Calculate boys' medal permutations: Calculate the number of ways to award medals to the boys. 12×11×10=132012 \times 11 \times 10 = 1320 ways to award the medals to the boys.
  3. Determine girls' medal permutations: Determine the number of ways to award medals to the girls.\newlineSince there are 66 girls competing and 33 medals to be awarded (gold, silver, and bronze), we need to calculate the number of permutations of 66 girls taken 33 at a time.\newlineThe formula for permutations is P(n,k)=n!(n−k)!P(n, k) = \frac{n!}{(n - k)!} where nn is the total number of items and kk is the number of items to choose.\newlineFor the girls, n=6n = 6 and k=3k = 3.\newlineP(6,3)=6!(6−3)!=6!3!=6×5×4P(6, 3) = \frac{6!}{(6 - 3)!} = \frac{6!}{3!} = 6 \times 5 \times 4
  4. Calculate girls' medal permutations: Calculate the number of ways to award medals to the girls.\newline6×5×4=1206 \times 5 \times 4 = 120 ways to award the medals to the girls.
  5. Determine total medal permutations: Determine the total number of ways to award all six medals.\newlineTo find the total number of ways to award all six medals, we multiply the number of ways to award the medals to the boys by the number of ways to award the medals to the girls.\newlineTotal number of ways == Number of ways for boys ×\times Number of ways for girls\newlineTotal number of ways =1320×120= 1320 \times 120
  6. Calculate total medal permutations: Calculate the total number of ways to award all six medals. 1320×120=158,4001320 \times 120 = 158,400 ways to award all six medals.

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