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  • In a group of 16 students, how many different ways can you select a team of 4 students to participate in a quiz if at least 2 of them must be from a specific grade?<stro

In a group of 16 students, how many different ways can you select a team of 4 students to participate in a quiz if at least 2 of them must be from a specific grade?

Option: 1

205


Option: 2

846


Option: 3

302


Option: 4

105


Answers (1)

best_answer

To solve this problem, we'll consider two cases: when exactly 2 students are chosen from the specific grade and when 3 or more students are chosen from the specific grade.

Case 1: Exactly 2 students from the specific grade are chosen.

In this case, we need to select the remaining 2 students from the remaining 16-2=14 students (excluding the 2 from the specific grade).

Number of ways to select 2 students from 14=14 C2

Case 2: 3 or more students from the specific grade are chosen.

In this case, we can choose 3 or 4 students from the specific grade. Let's consider each sub-case:

Sub-case 1: 3 students from the specific grade are chosen.

We need to select the remaining 1 student from the remaining 16-3=13 students.
Number of ways to select 1 student from 13=13 C1

Sub-case 2: All 4 students from the specific grade are chosen.

There is only 1 way to select all 4 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

Total number of ways =14 C 2+13 C 1+1

Now, let's calculate the value:

Total number of ways =91+13+1=105

Therefore, there are 105 different ways to select a team of 4 students to participate in the quiz, where at least 2 of them must be from a specific grade.

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Rishi

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