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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 14 boys and 10 girls are competing, how many different ways could the six medals possibly be given out? Answer Attempt 1 out of 2

Ask by Stephens Cummings. in the United States
Jan 22,2025

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Answer

There are 1,572,480 different ways to award the six medals.

Solution

To determine the number of different ways the six medals (gold, silver, bronze for both boys and girls) can be awarded in the competition, we'll break the problem down into two parts: awarding the medals to the boys and awarding them to the girls. ### 1. **Awarding Medals to the Boys** There are **14 boys** competing, and we need to assign three distinct medals: **gold**, **silver**, and **bronze**. - **Gold Medal:** Any of the 14 boys can win the gold medal. - **Silver Medal:** After awarding the gold medal, there are **13** remaining boys who can win the silver medal. - **Bronze Medal:** After awarding the gold and silver medals, there are **12** remaining boys who can win the bronze medal. **Total ways to award medals to boys:** \[ 14 \times 13 \times 12 = 2,184 \] ### 2. **Awarding Medals to the Girls** There are **10 girls** competing, and similarly, we need to assign three distinct medals: **gold**, **silver**, and **bronze**. - **Gold Medal:** Any of the 10 girls can win the gold medal. - **Silver Medal:** After awarding the gold medal, there are **9** remaining girls who can win the silver medal. - **Bronze Medal:** After awarding the gold and silver medals, there are **8** remaining girls who can win the bronze medal. **Total ways to award medals to girls:** \[ 10 \times 9 \times 8 = 720 \] ### 3. **Combining Both Results** Since the awarding of medals to boys and girls are independent events, the total number of different ways to award all six medals is the product of the two individual results. \[ \text{Total ways} = 2,184 \times 720 = 1,572,480 \] ### **Final Answer** There are **1,572,480** different ways to award the six medals.

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To find the number of different ways to award the medals, we first calculate the combinations for boys and girls separately and then multiply the results. For the boys, we need to select 3 winners from 14 competitors. The number of ways to select and arrange the top 3 can be calculated using permutations: - For gold, there are 14 choices. - For silver, there are 13 remaining choices after gold is awarded. - For bronze, there are 12 remaining choices after silver is awarded. So, for boys, it's \( 14 \times 13 \times 12 \). For the girls, the calculation is similar: - For gold, there are 10 choices. - For silver, 9 choices remaining. - For bronze, 8 choices remaining. So, for girls, it's \( 10 \times 9 \times 8 \). Now, we multiply the two results: - Total combinations for boys = \( 14 \times 13 \times 12 = 2184 \). - Total combinations for girls = \( 10 \times 9 \times 8 = 720 \). Therefore, the total combinations for assigning medals to both boys and girls becomes \( 2184 \times 720 = 1,572,768 \). So, the total ways the six medals could be awarded is 1,572,768.

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