A schoolteacher has to select the maximum possible number of different groups of 3 students out of a total of 6 students. In how many groups any particular student will be included?

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Q: 6 (IAS/2008)
A schoolteacher has to select the maximum possible number of different groups of 3 students out of a total of 6 students. In how many groups any particular student will be included?

question_subject: 

Maths

question_exam: 

IAS

stats: 

0,2,4,3,1,2,0

keywords: 

{'maximum possible number': [0, 0, 1, 0], 'different groups': [0, 0, 1, 0], 'many groups': [0, 0, 1, 0], 'students': [0, 1, 1, 1], 'particular student': [0, 0, 1, 0], 'schoolteacher': [0, 0, 1, 0]}

To determine the maximum possible number of different groups of 3 students out of a total of 6 students, we can use the concept of combinations.

The number of ways to choose a group of 3 students from a total of 6 can be calculated using the formula for combinations:

C(n, r) = n! / (r!(n - r)!)

where n is the total number of students (6 in this case) and r is the number of students we want to select (3 in this case).

Applying the formula:

C(6, 3) = 6! / (3!(6 - 3)!)

= (6 * 5 * 4) / (3 * 2 * 1)

= 20

So, there are 20 different groups of 3 students that can be formed from a total of 6 students.

Now, to determine how many groups any particular student will be included in, we can analyze the groups that each student can be a part of.

Let`s consider a specific student, Student A. We can calculate the number of groups that include Student A by selecting 2 students from the remaining 5 students (excluding Student A) to form a group of 3.

Using the formula for combinations again:

C(5, 2) = 5! / (2!(5 - 2)!)

= (5 * 4) / (2 * 1)

= 10

Therefore, any particular student (such as Student A) will be included in 10 different groups of 3 students.

Hence, the correct answer is Option 3: 10.