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  • In a group of 21 students, how many different ways can you select a team of 6 students to participate in a quiz if exactly 4 of them must be from a specific grade?<di

In a group of 21 students, how many different ways can you select a team of 6 students to participate in a quiz if exactly 4 of them must be from a specific grade?

Option: 1

29,652


Option: 2

74,896


Option: 3

10,520


Option: 4

18,496


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 the other 2 students are chosen from the remaining grades.

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

In this case, we need to select the remaining 2 students from the remaining 21 - 4 = 17 students (excluding the 4 from the specific grade).

Number of ways to select 2 students from 17=17\mathrm{C2}

Case 2: The other 2 students are chosen from the remaining grades.

In this case, we need to select 2 students from the remaining 21 - 4 = 17 students (excluding the 4 from the specific grade).

Number of ways to select 2 students from 17=\mathrm{17C2}

To calculate the total number of ways to form the team, we need to multiply the possibilities from both cases:

Total number of ways = Number of ways in Case 1 * Number of ways in Case 2

\mathrm{\text{Total number of ways = }\mathrm{17C2} \times17C2}

Now, let's calculate the value:

\mathrm{\text{Total number of ways = }\mathrm{136} \times136=18496}

Therefore, there are 18,496 different ways to select a team of 6 students to participate in the quiz, where exactly 4 of them must be from a specific grade.

Hence option 4 is correct.

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