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  • In a group of 17 students, how many different ways can you select a team of 5 students to participate in a quiz if at least 3 of them must be from a specific grade?<d

In a group of 17 students, how many different ways can you select a team of 5 students to participate in a quiz if at least 3 of them must be from a specific grade?

Option: 1

102


Option: 2

107


Option: 3

105


Option: 4

109


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 2 students from the remaining 17 - 3 = 14 students (excluding the 3 from the specific grade).

Number of ways to select 2 students from 14=14\mathrm{C}2

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

In this case, we can choose 4 or 5 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 1 student from the remaining 17 - 4 = 13 students.

Number of ways to select 1 student from 13=13\mathrm{C1}

Sub-case 2: 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

\mathrm{\text{Total number of ways =14C2+13C1+1 }}

Now, let's calculate the value:

\mathrm{\text{Total number of ways == 91 + 13 + 1 = 105 }}

Therefore, there are 105 different ways to select a team of 5 students to participate in the quiz, where at least 3 of them must be from a specific grade.

Hence option 3 is correct.

Posted by

Anam Khan

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