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Question

Question: In how many ways can \(7\) people be seated in a row on chairs if Jane and Joe must sit next to each...

In how many ways can 77 people be seated in a row on chairs if Jane and Joe must sit next to each other ?

Explanation

Solution

Try to solve this question by first finding the total ways to arrange two people. Once you get this idea try to find the seating arrangement of 7 people one at a time. Since we have been given a restriction, try to find the answer by naming the two of them as one person and then finding their internal arrangement as well.

Complete step by step answer:
First, try to find the answer to the question: if there are two people, say A, B, how can we arrange them in a row of two chairs in two ways. It can only be possible in two ways i.e. AB or BA. This is called permutation, which is the arrangement of objects in a certain way.If we were to be given 7 people, let us name them as A, B, C, D, E, F, G, the following will be their sitting arrangement.

The first chair can be occupied by any of those 7 people. The second chair can be occupied by one of 6 people because the first chair has been used by one person. In this pattern the third chair can be occupied by 5 people, the fourth chair by 4 people and so on.So the total possibilities from the above observation will be 7.6.5.4.3.2.1=7!=50407.6.5.4.3.2.1 = 7! = 5040 ways of arranging 7 people in a row respectively.

Now this cannot be our answer as there is a restriction given to us which is that Jane and Joe must sit next to each other. To solve this let us name A as Jane and B as Joe. Since we want these two together let us keep them together by naming them by letter X.Hence our number of people decreases to six that is X, C, D, E, F, G respectively.Hence referring from above the number of ways six people can be arranged in a row will be 6!=720.This also is not our last answer as a letter X which represents two people that is Jane and Joe can also be arranged in two ways i.e. AB or BA

Hence the total ways 7 people can be arranged in a row keeping Jane and Joe together will be 720×2=1440720 \times 2 = 1440 ways respectively. 720×2=1440720 \times 2 = 1440

Note: This kind of question can be solved by directly using the permutation formula if you are comfortable. But while doing this one has to be very particular about the arrangements and the conditions and restrictions given in the questions. Permutation is usually used when we are asked about arrangements about objects.