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  • A race consists of 10 participants, including 3 runners from Country A, 4 runners from Country B, and 3 runners from Country C. In how many different ways can the top 3 f

A race consists of 10 participants, including 3 runners from Country A, 4 runners from Country B, and 3 runners from Country C. In how many different ways can the top 3 finishers be from different countries?

 

Option: 1

320


Option: 2

218


Option: 3

612


Option: 4

462


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 3 runners from Country A, 4 runners from Country B, and 3 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(3,1)=3 }

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

The number of ways to select 1 runner from Country C is \mathrm{ C(3,1)=3 . }

Therefore, there are \mathrm{3^* 4 * 3=36} 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(3,1)=3 \text {. } }

The number of ways to select 1 runner from Country C is \mathrm{C(3,1)=3.}

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

Therefore, there are \mathrm{ 3 * 3 * 4=36} 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(4,1)=4.}

The number of ways to select 1 runner from Country A is \mathrm{ C(3,1)=3.}

The number of ways to select 1 runner from Country C is \mathrm{C(3,1)=3.}

Therefore, there are \mathrm{4 * 3 * 3=36} 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(4,1)=4.}

The number of ways to select 1 runner from Country C is \mathrm{C(3,1)=3.}

The number of ways to select 1 runner from Country A is \mathrm{C(3,1)=3.}

Therefore, there are \mathrm{4 * 3 * 3=36} 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(3,1)=3.}

The number of ways to select 1 runner from Country A is \mathrm{C(3,1)=3.}

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

Therefore, there are \mathrm{3^* 3^* 4=36} 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(3,1)=3.}

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

The number of ways to select 1 runner from Country A is \mathrm{C(3,1)=3.}

Therefore, there are \mathrm{ 3 \times 4 \times 3=36} 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{ 36+36+36+36+36+36=216 }

Therefore, there are 216 different ways the top 3 finishers in the race can be from different countries.

 

 

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shivangi.bhatnagar

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