20 passengers get on an airport shuttle. The shuttle route includes six hotels, and each passenger gets off the shuttle at his or her hotel. The driver records how many passengers leave the shuttle at each hotel. How many different possibilities exist?
At first glance, it appears that this is a standard combinations problem where we would use the notation 𝑛 choose 𝑟, where 𝑛 is the number of choices we have and 𝑟 is the number of times we’re choosing them. This is also written as 𝑛 over 𝑟 in large parentheses, which in turn is equal to 𝑛 factorial divided by 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial. We will discuss the factorial notation later.
When we’re dealing with any combinations problem however, it is important to know whether we’re dealing with or without replacement. This formula only works without replacement or without repetition, such as with the lottery. In this question, we’re not interested in which passengers get off at which hotel. We’re only interested in how we can organise the 20 passengers into the six hotels. We’re dealing with a problem with repetition and where order does not matter.
We’ll therefore use 𝑛 plus 𝑟 minus one over 𝑟. This is equal to 𝑛 plus 𝑟 minus one factorial over 𝑟 factorial multiplied by 𝑛 minus one factorial, where 𝑛 represent the total number of items — in this case, the number of hotels — and 𝑟 represent the number of passengers. There are six hotels. Therefore, 𝑛 is equal to six. And there are 20 passengers, so 𝑟 is equal to 20. Substituting in these values gives us six plus 20 minus one factorial divided by 20 factorial multiplied by six minus one factorial. This simplifies to 25 factorial divided by 20 factorial multiplied by five factorial.
We recall that the factorial of a number is the product of that integer and all smaller positive integers. For example, five factorial is equal to five multiplied by four multiplied by three multiplied by two multiplied by one. We could just type this into the calculator using the factorial button.
Let’s look at an alternative method by simplifying our expression. 25 factorial is equal to the product of all the integers between 25 and one. 20 factorial is equal to the product of all integers between 20 and one. As already shown, five factorial is the product of the integers between five and one. The integers 20 down to one appear on the numerator and denominator. Therefore, these can be cancelled. This leaves us with 25 multiplied by 24 multiplied by 23 multiplied by 22 multiplied by 21 on the top and five factorial on the bottom.
Whilst we could cancel this expression further as the numbers are quite high, we can type this into the calculator, giving us an answer of 53130. There are 53130 different possibilities or combinations that the 20 passengers could get off at the six hotels.