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- In a group of 30 students, how many different ways can you select a team of 8 students to participate in a quiz if exactly 4 of them must be from a specific grade?<stron
In a group of 30 students, how many different ways can you select a team of 8 students to participate in a quiz if exactly 4 of them must be from a specific grade?
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40250
12650
Answers (1)
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 4 students from the remaining 30-4=26 students (excluding the 4 from the specific grade).
Number of ways to select 4 students from 26=26 C4
Case 2: More than 4 students from the specific grade are chosen.
In this case, we can choose 5,6,7, or 8 students from the specific grade. Let's consider each subcase:
Sub-case 1: 5 students from the specific grade are chosen.
We need to select the remaining 3 students from the remaining 30-5=25 students.
Number of ways to select 3 students from 25=25 C3
Sub-case 2: 6 students from the specific grade are chosen.
We need to select the remaining 2 students from the remaining 30-6=24 students.
Number of ways to select 2 students from 24=24 C2
Sub-case 3:7 students from the specific grade are chosen.
We need to select the remaining 1 student from the remaining 30-7=23 students.
Number of ways to select 1 student from 23=23 C1
Sub-case 4: All 8 students from the specific grade are chosen.
There is only 1 way to select all 8 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+ Number of ways in Sub-case 4
Total number of ways =26 C4+25 C3+24 C 2+23 C1+1
Total number of ways =14,950+25,000+276+23+1=40,250
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