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  • In a group of 19 students, how many different ways can you select a team of 5 students to participate in a quiz if at least 2 of them must be from a specific grade?<d

In a group of 19 students, how many different ways can you select a team of 5 students to participate in a quiz if at least 2 of them must be from a specific grade?

Option: 1

816


Option: 2

865


Option: 3

846


Option: 4

874

 


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 more than 2 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 3 students from the remaining 19 - 2 = 17 students (excluding the 2 from the specific grade).

Number of ways to select 3 students from \mathrm{17=17 C 3 }

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

In this case, we can choose 3, 4, or 5 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 2 students from the remaining 19 - 3 = 16students.

Number of ways to select 2 students from \mathrm{16=16C2}

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

We need to select the remaining 1 student from the remaining 19 - 4 = 15 students.

Number of ways to select 1 student from \mathrm{15=15C1}

Sub-case 3: All 5 students from the specific grade are chosen.

There is only 1 way to select all 5 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 \mathrm{=17C3+16C2+15C1+1}

Now, let's calculate the value:

Total number of ways = 680 + 120 + 15 + 1 = 816

Therefore, there are 816 different ways to select a team of 5 students to participate in the quiz, where at least 2 of them must be from a specific grade.

Hence option 1 is correct.

 

 

 

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Riya

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