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  • In a group of 26 students, how many different ways can you select a team of 7 students to participate in a quiz if at least 4 of them must be from a specific grade?

In a group of 26 students, how many different ways can you select a team of 7 students to participate in a quiz if at least 4 of them must be from a specific grade?

 

 

Option: 1

1221


 


Option: 2

1771


Option: 3

1881

 


Option: 4

1661


Answers (1)

best_answer

To solve this problem, we'll consider two cases: when exactly 4 students are chosen from the specific grade and when more than 4 students are chosen from the specific grade.

Case 1: Exactly 4 students from the specific grade are chosen.

In this case, we need to select the remaining 3 students from the remaining 26 - 4 = 22 students (excluding the 4 from the specific grade).

Number of ways to select 3 students from22=22\mathrm{C}3

Case 2: More than 4 students from the specific grade are chosen.

In this case, we can choose 5, 6, or 7 students from the specific grade. Let's consider each sub-case:

Sub-case 1: 5 students from the specific grade are chosen.

We need to select the remaining 2 students from the remaining 26 - 5 = 21 students.

Number of ways to select 2 students from \mathrm{21=21C2}

Sub-case 2: 6 students from the specific grade are chosen.

We need to select the remaining 1 student from the remaining \mathrm{26 - 6 = 20 }students.

Number of ways to select 1 student from

Sub-case 3: All 7 students from the specific grade are chosen.

There is only 1 way to select all 7 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

\mathrm{\text{Total number of ways =22C3+21C2+20C1+1} }

Now, let's calculate the value:

\mathrm{\text{Total number of ways = 1540 + 210 + 20 + 1 = 1771} }

Therefore, there are 1771 different ways to select a team of 7 students to participate in the quiz, where at least 4 of them must be from a specific grade.

Hence option 2 is correct.

Posted by

vinayak

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