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- In a group of 25 students, how many different ways can you select a team of 7 students to participate in a quiz if at least 4 of thern must be from a specific grade?<str
In a group of 25 students, how many different ways can you select a team of 7 students to participate in a quiz if at least 4 of thern must be from a specific grade?
9632
8416
1540
8200
Answers (1)
To solve this problem, we'll consider two cases: when exactly 4 students are chosen from the specific grade and when 5 or more 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 25-4=21 students (excluding the 4 from the specific grade).
Number of ways to select 3 students from 21= 21 C3
Case 2: 5 or more 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 25-5=20 students.
Number of ways to select 2 students from 20=20 C2
Sub-case 2: 6 students from the specific grade are chosen.
We need to select the remaining 1 student from the remaining 25-6=19 students.
Number of ways to select 1 student from 19=19 C1
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
Total number of ways =21 C 3+20 C 2+19 C 1+1
Now, let's calculate the value:
Total number of ways =1330+190+19+1=1540
Therefore, there are 1540 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.
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