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  • A race consists of 12 participants, including 4 runners from Country A, 6 runners from Country B, and 2 runners from Country C. In how many different ways can the top 3 f

A race consists of 12 participants, including 4 runners from Country A, 6 runners from Country B, and 2 runners from Country C. In how many different ways can the top 3 finishers be from different countries?

 

Option: 1

366


Option: 2

288


Option: 3

188


Option: 4

488


Answers (1)

best_answer

To calculate the number of different ways the top 3 finishers in the race can be from different countries, we can consider the possible combinations of countries for the top 3 positions.

Since there are 4 runners from Country A, 6 runners from Country B, and 2 runners from Country C, we have the following possibilities:

Country A for the 1st position, Country B for the 2nd position, and Country C for the 3rd position.

The number of ways to select 1 runner from Country A is \mathrm{C(4,1)=4.}

The number of ways to select 1 runner from Country B is \mathrm{C(6,1)=6.}

The number of ways to select 1 runner from Country C is \mathrm{C(2,1)=2.}

Therefore, there are \mathrm{4 * 6 * 2=48} possible combinations for this case.

Country A for the 1st position, Country C for the 2nd position, and Country B for the 3rd position.

The number of ways to select 1 runner from Country A is \mathrm{ C(4,1)=4 }

The number of ways to select 1 runner from Country C is \mathrm{C(2,1)=2.}

The number of ways to select 1 runner from Country B is \mathrm{C(6,1)=6.}

Therefore, there are \mathrm{4 * 2 * 6=48} possible combinations for this case.

Country B for the 1st position, Country A for the 2nd position, and Country C for the 3rd position.

The number of ways to select 1 runner from Country B is \mathrm{C(6,1)=6.}

The number of ways to select 1 runner from Country A is \mathrm{C(4,1)=4.}

The number of ways to select 1 runner from Country C is \mathrm{C(2,1)=2.}

Therefore, there are \mathrm{6 * 4 * 2=48} possible combinations for this case.

Country B for the 1st position, Country C for the 2nd position, and Country A for the 3rd position.

The number of ways to select 1 runner from Country B is \mathrm{C(6,1)=6.}

The number of ways to select 1 runner from Country C is \mathrm{C(2,1)=2.}

The number of ways to select 1 runner from Country A is \mathrm{C(4,1)=4.}

Therefore, there are \mathrm{6 * 2 * 4=48} possible combinations for this case.

Country C for the 1st position, Country A for the 2nd position, and Country B for the 3rd position.

The number of ways to select 1 runner from Country C is \mathrm{C(2,1)=2.}

The number of ways to select 1 runner from Country A is \mathrm{C(4,1)=4.}

The number of ways to select 1 runner from Country B is \mathrm{C(6,1)=6.}

Therefore, there are \mathrm{2 * 4 * 6=48} possible combinations for this case.

Country C for the 1st position, Country B for the 2nd position, and Country A for the 3rd position.

The number of ways to select 1 runner from Country C is \mathrm{C(2,1)=2.}

The number of ways to select 1 runner from Country B is \mathrm{C(6,1)=6.}

The number of ways to select 1 runner from Country A is \mathrm{C(4,1)=4.}

Therefore, there are \mathrm{2 \times 6 \times 4=48} possible combinations for this case.

To find the total number of different ways the top 3 finishers can be from different countries, we sum up the possibilities from each case:

\mathrm{ 48+48+48+48+48+48=288 . }

Therefore, there are 288 different ways the top 3 finishers in the race can be from different countries.

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vishal kumar

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