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  • A team of 5 players needs to be selected from a group of 9 athletes. However, two specific athletes, Jack and Olivia, cannot be on the team together. In how many ways can the team be formed?<di

A team of 5 players needs to be selected from a group of 9 athletes. However, two specific athletes, Jack and Olivia, cannot be on the team together. In how many ways can the team be formed?

Option: 1

52


Option: 2

84


Option: 3

70


Option: 4

12


Answers (1)

best_answer

To calculate the number of ways the team can be formed, we need to consider two scenarios: Jack is on the team and Olivia is not, and Olivia is on the team and Jack is not.

Scenario 1: Jack is on the team and Olivia is not:

In this case, we need to select 4 more players from the remaining 7 athletes (excluding Jack and Olivia). The number of ways to do this is given by the combination formula: \mathrm{C(7,4)=7 ! /(4 ! \times(7-4) !)=35 \text {. }}

Scenario 2: Olivia is on the team and Jack is not:

Similar to Scenario 1, we need to select 4 more players from the remaining 7 athletes (excluding Jack and Olivia). Again, the number of ways to do this is given by the combination formula: \mathrm{C(7,4)=7 ! /(4 ! \times(7-4) !)=35 \text {. }}

Since these two scenarios are mutually exclusive (Jack and Olivia cannot be on the team together), we can simply add the results:

\mathrm{Number \: of \: ways = Scenario 1 + Scenario 2 = 35 + 35 = 70.}

Therefore, there are 70 ways to form a team of 5 players, given that Jack and Olivia cannot be on the team together.

Hence option 3 is correct.

 

 

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Rishi

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