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  • In a group of 24 students, how many different ways can you select a team of 6 students to participate in a quiz if at least 3 of them must be from a specific grade?<stro

In a group of 24 students, how many different ways can you select a team of 6 students to participate in a quiz if at least 3 of them must be from a specific grade?

Option: 1

2030


Option: 2

4020


Option: 3

1050


Option: 4

1540


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 24-3=21 students (excluding the 3 from the specific grade).

Number of ways to select 3 students from 21=21 C3

Case 2: More than 3 students from 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 24-4=20 students.

Number of ways to select 2 students from 20=20 C2

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

We need to select the remaining 1 student from the remaining 24 - 5=19 students.

Number of ways to select 1 student from 19=19 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 =21 C3+20 C2+19 C1+1

Now, let's calculate the value:

Total number of ways =1330+190+19+1=1540

Therefore, there are 1540 different ways to select a team of 6 students to participate in the quiz, where at least 3 of them must be from a specific grade.

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