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?

(e) Using the results of parts (a) through (d), write a combinatorial identity.

Short answer

(a) The possible no. of choices are$\left(\begin{array}{c}3n\\ 3\end{array}\right)$.

(b) The possible no. of choices that all $3$students are in the same class are$3\left(\begin{array}{c}n\\ 3\end{array}\right)$.

(c) The possible no. of choices that $2$of the $3$students are in the same class and the other student is in a different class are$3\times \left(\begin{array}{c}n\\ 2\end{array}\right)\times 2\times \left(\begin{array}{c}n\\ 1\end{array}\right)$.

(d) The possible no. of choices that all $3$students are in different classes are$n^3$

(e) Using the results of parts (a) through (d), the combinatorial identity will berole="math" localid="1649084344661" $\left(\begin{array}{c}3n\\ 3\end{array}\right)=3\left(\begin{array}{c}n\\ 3\end{array}\right)+3\left(\begin{array}{c}n\\ 2\end{array}\right)\times 2\left(\begin{array}{c}n\\ 1\end{array}\right)+n^3$

Step-by-step solution

Part (a) Step 1. Given information.

It is given that, there are three classes and each class has $n$number of students and we have to select $3$students from the total number of students.

Part (a) Step 2. Find the possible no. of choices.

To select$3$students from$3n$students, the possible no. of choices arelocalid="1649084012796" $\left(\begin{array}{c}3n\\ 3\end{array}\right)$.

Part (b) Step 1. Find the possible no. of choices that all 3 students are in the same class.

No. of ways of selecting $3$students from a class of $n$students is role="math" localid="1649055941717" $=\left(\begin{array}{c}n\\ 3\end{array}\right)$.

No. of ways of choosing any of the $3$classes $=\left(\begin{array}{c}3\\ 1\end{array}\right)=3$.

Therefore, the possible no. of choices that all $3$ students are in the same class are$3\left(\begin{array}{c}n\\ 3\end{array}\right)$.

Part (c) Step 1. Find the possible no. of choices that 2 of the 3 students are in the same class and the other student is in a different class.

No. of ways of selecting $2$students from a class of $n$students is $=\left(\begin{array}{c}n\\ 2\end{array}\right)$.

No. of ways of selecting $1$student from a class of $n$students is $=\left(\begin{array}{c}n\\ 1\end{array}\right)$.

The class from which $2$students are selected can be chosen in $\left(\begin{array}{c}3\\ 1\end{array}\right)\text{ways}=3$

The class from which $1$students is selected can be chosen in $\left(\begin{array}{c}2\\ 1\end{array}\right)\text{ways}=2$

Therefore, the possible no. of ways are$=3\times \left(\begin{array}{c}n\\ 2\end{array}\right)\times 2\times \left(\begin{array}{c}n\\ 1\end{array}\right)$

Part (d) Step 1. Find the possible no. of choices that all 3 students are in different classes.

No. of ways of selecting$1$student from $n$students is $\left(\begin{array}{c}n\\ 1\end{array}\right)=n$

No. of choices for $3$students $=n\times n\times n$

Therefore, the possible no. of choices that all 3 students are in different classes$=n^3$.

Part (e) Step 1. Write a combinatorial identity.

Using the results of parts (a) through (d), the combinatorial identity will be

$\left(\begin{array}{c}3n\\ 3\end{array}\right)=3\left(\begin{array}{c}n\\ 3\end{array}\right)+3\left(\begin{array}{c}n\\ 2\end{array}\right)\times 2\left(\begin{array}{c}n\\ 1\end{array}\right)+n^3$