Question

A pod of $6$ dolphins always swim single file, with $3$ females at the front and $3$ males in the rear. In how many ways different arrangements can the dolphin swim?
A. $24$
B. $36$
C. $30$
D. $18$

Answer 1

Number of ways in which $3$ male dolphins can swim$= 3!$
Similarly number of ways in which $3$ dolphins can swim$= 3!$
Total number of arrangements is the product of both the cases.

Complete step by step solution:
According to the question, there is a pod of $6$ dolphins which means the group of dolphins that always swim single file, with $3$ females at the front and $3$ males in the rear. So there are a total of $6$ dolphins out of which three are male and three are females. Now we need to find the number of arrangements possible for such an event to happen. So firstly we need to find the total number of ways in which female dolphins can swim and similarly we also need to find the number of ways in which the male dolphins can swim and after that their product will give us the total number of arrangements possible for the given case.
As we know that the arrangement of $n$ different things can be done in $n!$ ways.
So the number of ways in which $3$female dolphins can swim$= 3!$
Similarly male dolphins can also swim in $3!$ ways
And we should also know what factorial means $3! = (3)(2)(1) = 6$
So both the male and the female dolphins can swim in $6$ different ways possible.
Total number of arrangements is the product of both the cases.

So we get the total possible ways$= (6)(6) = 36{\text{ ways}}$

Note:
For arranging $n$ different things in the circular arrangements, the number of ways possible are $(n - 1)!$.