If 13𝑃𝑟 is equal to 1,235,520, find 𝑟 plus two factorial.
In this question, we are given an expression using permutation notation 𝑛𝑃𝑟. This is the number of different ways to order our objects from 𝑛 total distinct objects. And 𝑛𝑃𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial. The value of 𝑛 in our expression is 13. This means that 13 factorial divided by 13 minus 𝑟 factorial is equal to 1,235,520. We can rearrange this equation such that 13 minus 𝑟 factorial is equal to 13 factorial divided by 1,235,520. Typing the left-hand side of our equation into our calculator gives us 5,040. And we know this must be equal to 13 minus 𝑟 factorial.
It may not be obvious what integer value is equal to 13 minus 𝑟. However, we recall that five factorial is equal to 120. Multiplying this by six, we see that six factorial is 720. Since 720 multiplied by seven is 5,040, this is equal to seven factorial. We have seven factorial is equal to 13 minus 𝑟 factorial. This means that seven must be equal to 13 minus 𝑟 and 𝑟 is therefore equal to 13 minus seven, which equals six. 13𝑃 six is equal to 1,235,520.
This isn’t the final answer though, as we are asked to find 𝑟 plus two factorial. Since 𝑟 is equal to six, this is equal to eight factorial. Since seven factorial is 5,040, we can multiply this by eight giving us 40,320, which is our value of eight factorial.
If 13𝑃𝑟 is equal to 1,235,520, then 𝑟 plus two factorial is 40,320.