Question

Find the number of ways that 5 keys can be put in a ring?
(a) $\dfrac{4!}{2}$
(b) $\dfrac{5!}{2}$
(c) $4!$
(d) $5!$

Answer 1

Hint: We start solving the problem by assuming that the clockwise and anticlockwise order of arrangements are considered the same. We then recall the fact that the number of ways of arranging n different objects in a circle if clockwise and anticlockwise order of arrangements is considered the same is $\dfrac{\left( n-1 \right)!}{2}$ to proceed through the problem. We then use this fact and make necessary calculations to get the required answer.

Complete step by step solution:
According to the problem, we are asked to find the total number of ways that 5 keys can be put in a ring.
Let us assume that the clockwise and anticlockwise order of arrangements are considered the same.
We know that the number of ways of arranging n different objects in a circle if clockwise and anticlockwise order of arrangements is considered the same is $\dfrac{\left( n-1 \right)!}{2}$.
So, the number of ways of arranging 5 keys in a ring is $\dfrac{\left( 5-1 \right)!}{2}=\dfrac{4!}{2}$.
We have found the number of ways to arrange the given 5 keys in a ring as $\dfrac{4!}{2}$ ways.
So, the correct answer is “Option A”.

Note: Whenever we get this type of problems, we first check whether the order of arrangement is considered same or different. If the order of arrangement is considered different then the obtained answer will vary. We can also solve this problem as shown below:
Let us assume that the numbers on keeps be 1, 2, 3, 4 and 5.
Let us the arranging the keys in the ring are as shown below:

We can arrange all keys in $5!$ ways. But the following ways also represent the previous way of arranging the keys.

So, we need to divide $5!$ with 5 to get the effective number of ways to fill.
So, we get $\dfrac{5!}{5}=4!$.
We have assumed that clock wise and anti clock wise order of arrangements is considered as same. Which means that the following two ways of arranging keys is considered as same.

So, we need to divide $4!$ with 2 to get the required answer which is $\dfrac{4!}{2}$.