Question

Consider three classes, each consisting of n students. From this group of 3n students, a group of 3 students is to be chosen.

a. How many choices are possible?

b. How many choices are there in which all 3 students are in the same class?

c. How many choices are there in which 2 of the 3 students are in the same class and the other student is in a different class.

d. How many choices are there in which all 3 students are in different classes?

Answer 1

Answer:

Step-by-step explanation:

Given that there are three classes, each consisting of n students. From this group of 3n students, a group of 3 students is to be chosen.

a) Out of 3n students to draw 3 students we use combination since order does not matter.

Hence no of ways = $3nCn$

b) If three students are to be in same class, either from I class or Ii or III

No of ways = nC3 + nc3+nc3 = 3(nC3)

c) 2 of the 3 students are in the same class and the other student is in a different class.

2 can be either from I or II or III and the remaining from any one of other classes.

So no of ways = $3(nC2) + 2nC1$

d) in which all 3 students are in different classes

Each student 1 is selected from each class of n students

So $3(nC1) = 3n$

Answer 2

The ways of selecting the students is an illustration of combination.

• The total number of choice is $\mathbf{ ^{3n}C_3}$.
• The total number of choices, where the 3 students are in the same class is $\mathbf{ 3^{n}C_3}$
• The total number of choices, where 2 of 3 students are in the same class is $\mathbf{3 \times ^{n}C_2 \times ^{2n}C_1}$.
• The total number of choices, all students are in different class is $\mathbf{3 ^{n}C_1 }$

The given parameters are:

$\mathbf{Group = 3n}$

$\mathbf{Students = 3}$

(a) The total number of choices

This means that 3 students are to be selected from 3n students.

So, the number of choice is:

$\mathbf{Choice = ^{3n}C_3}$

(b) The total number of choices, where the 3 students are in the same class

The number of classes is 3.

Each of the class, has n students

So, this means that 3 students are to be selected from n students in each class.

So, the number of choice is:

$\mathbf{Choice = 3 \times ^{n}C_3}$

$\mathbf{Choice = 3^{n}C_3}$

(c) The total number of choices, where 2 of 3 students are in the same class

Using the same analysis as (b), we have the number of classes to be 3.

Each of the class, has n students

So, this means that:

• 2 students are to be selected from n students in each class.
• The last student is selected from the remaining 2 classes.

So, the number of choice is:

$\mathbf{Choice = 3 \times ^{n}C_2 \times ^{2n}C_1}$

(d) The total number of choices, all students are in different class

Each class has n students

So, this means that: 1 student is selected from each class

So, the number of choice is:

$\mathbf{Choice = 3 \times ^{n}C_1 }$

$\mathbf{Choice = 3 ^{n}C_1 }$

Hence, the total number of choices, all students are in different class is $\mathbf{3 ^{n}C_1 }$

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