In a town, there are nine hotels. In how many ways can three tourists stay in the town, given that they each want to stay in a different hotel?
In order to answer this question, we need to use our knowledge of permutations. A permutation is an arrangement of a collection of items without repetition and where order matters. The notation for this is 𝑛P𝑟, where 𝑟 is the number of items we are selecting and 𝑛 is the total number of items. We can calculate this using the formula 𝑛 factorial divided by 𝑛 minus 𝑟 factorial.
In this question, there are nine hotels. Therefore, 𝑛 is equal to nine. There are three tourists, so 𝑟 is equal to three. We need to calculate nine P three, which is equal to nine factorial divided by six factorial as nine minus three is equal to six. We know that 𝑛 factorial can be rewritten as 𝑛 multiplied by 𝑛 minus one factorial. This means that nine factorial is equal to nine multiplied by eight multiplied by seven multiplied by six factorial.
We can then divide the numerator and denominator by six factorial. This leaves us with nine multiplied by eight multiplied by seven. Nine multiplied by eight is 72. And multiplying this by seven gives us 504. There are 504 ways that the three tourists can stay in the town across the nine hotels.
We could also have worked out the answer on a scientific calculator by simply typing nine, the 𝑛P𝑟 button, three, and then the equals button. This would also give us an answer of 504.