Video Transcript
Five 𝑃 𝑟 over five 𝑃 𝑟 minus one is equal to blank.
If we know the definition of permutations in the form 𝑛 𝑃 𝑟, they are 𝑛 factorial over 𝑛 minus 𝑟 factorial, which means the numerator of this ratio can be rewritten as five factorial over five minus 𝑟 factorial. In the denominator, the 𝑟 position is 𝑟 minus one. And so we should write that as five factorial over five minus 𝑟 minus one. We’re taking the numerator and dividing it by the denominator. We know that when we’re dividing fractions, that’s the same as multiplying by the reciprocal, so we’ll be multiplying by five minus 𝑟 minus one factorial over five factorial. And then we have a five factorial in the numerator and the denominator that cancel out.
At this point, it might seem like there’s not much else we can do, so let’s turn our attention to the numerator for a second. We can distribute this negative to both the 𝑟 and to the one, which gives us five minus 𝑟 plus one factorial. And we can combine that five and one so that we have six minus 𝑟 factorial over five minus 𝑟 factorial. From there, we’ll need to use the property of factorials that 𝑛 factorial is equal to 𝑛 times 𝑛 minus one factorial. In this case, the 𝑛 position in our numerator is six minus 𝑟. And that means we could say that six minus 𝑟 factorial is equal to six minus 𝑟 times six minus 𝑟 minus one factorial.
What we notice here is that we’ll have six minus one, which we can simplify to five so that this term becomes five minus 𝑟 factorial, which means we have a five minus 𝑟 factorial term in the numerator and the denominator. And it means the only remaining term is six minus 𝑟, so five 𝑃 𝑟 over five 𝑃 𝑟 minus one will be equal to six minus 𝑟.
