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  • In a group of 27 students, how many different ways can you select a team of 7 students to participate in a quiz if at least 5 of them must be from a specific grade?

In a group of 27 students, how many different ways can you select a team of 7 students to participate in a quiz if at least 5 of them must be from a specific grade?

 

 

Option: 1

352


Option: 2

153


Option: 3

253


Option: 4

128


Answers (1)

best_answer

To solve this problem, we'll consider two cases: when exactly 5 students are chosen from the specific grade and when more than 5 students are chosen from the specific grade.

Case 1: Exactly 5 students from the specific grade are chosen.

In this case, we need to select the remaining 2 students from the remaining 27 - 5 = 22 students (excluding the 5 from the specific grade).

Number of ways to select 2 students from 22=22\mathrm{C}2

Case 2: More than 5 students from the specific grade are chosen.

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

Sub-case 1: 6 students from the specific grade are chosen.

We need to select the remaining 1 student from the remaining 27 - 6 = 21 students.

Number of ways to select 1 student from 21=21\mathrm{C}1

Sub-case 2: 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:

\mathrm{\text{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 =22C2+21C1+1 }}

Now, let's calculate the value:

\mathrm{\text{Total number of ways =231 + 21 + 1 = 253 }}

Therefore, there are 253 different ways to select a team of 7 students to participate in the quiz, where at least 5 of them must be from a specific grade.

Hence option 3 is correct.

Posted by

Suraj Bhandari

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