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  • In a group of 22 students, how many different ways can you select a team of 6 students to participate in a quiz if exactly 3 of them must be from a specific grade? .<div class='qna-optio

In a group of 22 students, how many different ways can you select a team of 6 students to participate in a quiz if exactly 3 of them must be from a specific grade?
.

Option: 1

2920


Option: 2

1140


Option: 3

7452


Option: 4

1200


Answers (1)

best_answer

To solve this problem, we'll consider two cases: when exactly 3 students are chosen from the specific grade and when more than 3 students are chosen from the specific grade.

Case 1: Exactly 3 students from the specific grade are chosen.

In this case, we need to select the remaining 3 students from the remaining 22-3=19 students (excluding the 3 from the specific grade).

Number of ways to select 3 students from 19=19 C3

Case 2: More than 3 students frorn the specific grade are chosen.

In this case, we can choose 4, 5, or 6 students from the specific grade. Let's consider each sub-case:

Sub-case 1: 4 students from the specific grade are chosen.

We need to select the remaining 2 students from the remaining 22 - 4=18 students.

Number of ways to select 2 students from 18=18 C2

Sub-case 2: 5 students from the specific grade are chosen.

We need to select the remaining 1 student from the remaining 22 - 5=17 students.

Number of ways to select 1 student from 17=17 C1

Sub-case 3: All 6 students from the specific grade are chosen.

There is only 1 way to select all 6 students from the specific grade.

To calculate the total number of ways to form the team, we need to sum up the possibilities from both cases and all sub-cases:

Total number of ways = Number of ways in Case 1+ Number of ways in Sub-case 1+ Number of ways in Sub-case 2+ Number of ways in Sub-case 3

Total number of ways =19 C3+18 C2+17 C1+1

Now, let's calculate the value:

Total number of ways =969+153+17+1=1140

Therefore, there are 1140 different ways to select a team of 6 students to participate in the quiz, where exactly 3 of them must be from a specific grade.

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rishi.raj

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