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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? Answer:

In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If 66 boys and 1515 girls are competing, how many different ways could the six medals possibly be given out?\newlineAnswer:

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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 66 boys and 1515 girls are competing, how many different ways could the six medals possibly be given out?\newlineAnswer:
  1. Calculate Boys Medals: Calculate the number of ways to give out medals to the boys.\newlineSince there are 66 boys and 33 medals (gold, silver, bronze), we use permutations because the order matters (gold is different from silver, which is different from bronze).\newlineNumber of ways to give out medals to the boys = 6P3=6!(6−3)!6P3 = \frac{6!}{(6-3)!}
  2. Calculate Boys Medals Result: Perform the calculation for the number of ways to give out medals to the boys.\newline6!/3!=(6×5×4×3×2×1)/(3×2×1)=6×5×4=1206! / 3! = (6 \times 5 \times 4 \times 3 \times 2 \times 1) / (3 \times 2 \times 1) = 6 \times 5 \times 4 = 120
  3. Calculate Girls Medals: Calculate the number of ways to give out medals to the girls.\newlineSince there are 1515 girls and 33 medals, we again use permutations.\newlineNumber of ways to give out medals to the girls = 15P3=15!(15−3)!15P3 = \frac{15!}{(15-3)!}
  4. Calculate Girls Medals Result: Perform the calculation for the number of ways to give out medals to the girls. 15!/12!=(15×14×13×12×11×…×1)/(12×11×…×1)=15×14×13=273015! / 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
  5. Calculate Total Ways: Calculate the total number of different ways the six medals can be given out.\newlineSince 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.\newlineTotal number of ways == Number of ways for boys ×\times Number of ways for girls
  6. Calculate Total Ways Result: Perform the final calculation to find the total number of different ways the six medals can be given out.\newlineTotal number of ways = 120120 (for boys) ×\times 27302730 (for girls) = 327600327600

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