Determine the number of different ways for four players to sit on 11 seats in a row.
We want to find the number of unique positions that these four players could arrange themselves in on 11 seats. We can approach this problem in two ways, firstly, using a logical perspective. We can consider the number of choices that each player has. The first player can sit wherever they like, so there are 11 seats for them to choose from. But when the second player comes along, they find that one seat is already filled, so they only have 10 seats to choose from. Player three can only choose from the remaining nine seats. And finally, player four has only eight seats to choose from.
Any of player one’s choices can be combined with any of player’s two choices and any of player three’s and any of player four’s. So the total number of different ways that these four players can sit on the 11 seats is 11 multiplied by 10 multiplied by nine multiplied by eight. That’s 110 multiplied by 72, which is 7,920.
The other way we could approach this problem is to recognize that what we’re being asked to calculate is a permutation, the number of ways we can select four things by which we mean the seats these players sit in from 11 when the order matters.
It’s important that we recognize that the order does matter here. Person one sitting in seat three is considered different from person four sitting in seat three. We can use the notation 11𝑃 four to denote the number of ways of choosing and then arranging four distinct things from a group of 11 distinct things. In general, the notation 𝑛𝑃𝑟 means the number of permutations of 𝑟 distinct objects from 𝑛 distinct objects. And it’s calculated as 𝑛 factorial over 𝑛 minus 𝑟 factorial, which you may also see using the notation of an exclamation mark.
11𝑃 four then is 11 factorial over seven factorial. That’s the product of the integers from one to 11 over the product of the integers from one to seven. Of course, large parts of this calculation cancel out, leaving just 11 multiplied by 10 multiplied by nine multiplied by eight in the numerator and one in the denominator. That’s the same as the calculation we found using our logical approach. So we know that the answer will once again be 7,920.
If you have a calculator, you can also evaluate this using the 𝑛𝑃𝑟 button. The location of this will be specific to your calculator. But on mine, it is the second function above the multiplication button. So I have to press shift and then this button in order to access this function.
We found then that there are 7,920 ways for four players to sit on 11 seats in a row.