How many ways can two people sit on eight chairs?
So this question is a permutations question. And to help us deal with permutations, we have an equation. So we can say that the number of permutations of our objects chosen from a set of 𝑛 is gonna be given by. We’ve got 𝑛 permute 𝑟 is equal to 𝑛 factorial over 𝑛 minus 𝑟 factorial.
So just to remind us what some of the notation means, if we’ve got 𝑛 factorial, what this is is the product of all positive integers less than or equal to 𝑛. For example, five factorial is equal to five multiplied by four multiplied by three multiplied by two multiplied by one.
So if we take a look at the question, we’ve got our 𝑟 is equal to two. So our number of objects is two, cause there are two people. And our 𝑛 is gonna be equal to eight, because there are eight chairs for them to sit in. So we can write eight permute two is going to be equal to eight factorial divided by eight minus two factorial.
So at this point, you think, well, just put it straight into a calculator. But actually, we can cancel this down to solve the problem easily. And that’s because we can rewrite the numerator as eight multiplied by seven multiplied by six factorial. Because eight factorial would be eight multiplied by seven multiplied by six multiplied by five, et cetera.
So therefore, if six factorial is six multiplied by five multiplied by four, et cetera, then we can rewrite it in this way. And then on the denominator, we’ve just got six factorial. That’s cause eight minus two is six. So therefore, if we divide the numerator and denominator by six factorial, these will cancel. So we’re just left with eight multiplied by seven, which is gonna give us an answer of 56. So therefore, we can say that there are 56 ways that two people can sit on eight chairs.