In how many ways can 7 people be seated in a row o chairs if Jane and Joe must sit next to each other?
Seven people can be seated in
This is a question about permutations. A permutation is an arrangement of a certain amount of objects in a specific order.
For example, if we have two people,
Each of these two arrangements is a permutation. Thus, there are only two possible permutations in the example.
If we have seven people it gets a bit trickier. We must use the multiplication principle.
If we have seven people and we want to know how many ways we can arrange them in a row of seven chairs, using the multiplication principle, we multiply all the options together to get the total number of arrangements. E.g. there are 7 options for the first seat, 6 for the second (because one has been used), 5 for the third, 4 for the fourth, and so on.
So there are 5040 ways of arranging seven people in a row of seven chairs.
I think the proof of this is best seen by making a tree diagram and noting that the number of branches at the end tells you the number of possible arrangements/permutations. The tree for seven people is way too big so if you want to test this, do it with a smaller number, such as three people (there will be
Now, let's call each person
The answer is not 5040 because we have a restriction, two of the people must sit next to each other. We'll call Jane and Joe,
Now we effectively have six objects
Using the multiplication principle, we can arrange six objects in 6! ways
But we also have to take into consideration the group,
To take this into account and get the total number of arrangements, we must use the multiplication principle again. This involves multiplying the number of arrangements of six objects by the number of arrangements of the group,
This makes sense because we have 720 arrangements where