Question Video: Using the Properties of Permutations to Find the Value of an Unknowns and Evaluate a Factorial Expression Mathematics

Video Transcript

If 49𝑃 𝑟 plus three equals 34 times 49𝑃 𝑟 plus two, find the value of 𝑟 minus six factorial.

One way to solve this will be to first convert each of these permutations into factorial form using the definition 𝑛𝑃𝑟 equals 𝑛 factorial over 𝑛 minus 𝑟 factorial. We rewrite 49𝑃 𝑟 plus three as 49 factorial over 49 minus 𝑟 plus three factorial. Be careful to notice that in this case, our 𝑟, the number of elements we’re choosing, is 𝑟 plus three and that the subtraction applies to both the 𝑟 and the three. The other side of the equation becomes 34 times 49 factorial over 49 minus 𝑟 plus two factorial. We have 49 factorial in the numerator on both sides of the equation, which means we can multiply both sides of the equation by one over 49 factorial.

On the left, we’ll be left with the numerator of one. And in the denominator, we can subtract 𝑟 and subtract three from 49, which will give us one over 46 minus 𝑟 factorial. The numerator on the right side becomes 34, and the denominator is 47 minus 𝑟 factorial. At this point, it might not seem clear what we should do. But since we’re trying to solve for 𝑟, it’s a good idea to get both 𝑟-values on the same side of the equation. So we’ll multiply both sides of the equation by 47 minus 𝑟 factorial. This gives us 47 minus 𝑟 factorial over 46 minus 𝑟 factorial equals 34. Our denominator is a factorial that is one less than our numerator.

We know that 𝑛 factorial is equal to 𝑛 times 𝑛 minus one factorial. We can apply this here, but we need to be careful. In our case, the 𝑛 is 47 minus 𝑟. And that means 47 minus 𝑟 factorial is equal to 47 minus 𝑟 times 47 minus 𝑟 minus one factorial. We can simplify 47 minus 𝑟 factorial to be equal to 47 minus 𝑟 times 46 minus 𝑟 factorial. If we substitute this in for our numerator, the 46 minus 𝑟 factorial cancels out in the numerator and the denominator. We then have 47 minus 𝑟 equals 34. If we subtract 47 from both sides, we find that negative 𝑟 equals negative 13, which makes 𝑟 equal to positive 13. Our final step will be to calculate 𝑟 minus six factorial, which we now know is 13 minus six factorial. Solving this by hand or with a calculator will yield that seven factorial equals 5,040.