# full unmatch probability [closed]

You have 5 distinct color envelops and 5 distinct color letters. You randomly put one letter in each envelop. What is the probability of no pair matching in color?

I can find the answer with brute force solving the complementary problem and then subtracting from 1. Looking for an elegant way to think about it, if it exists.

• Search term: derangements – RobPratt May 21 at 12:17
• thanks @RobPratt I did look at it and it seems similar to inclusion exclusion principle. Thanks for pointing it out. I am just wondering if there is a 2-3 line argument to solve this, maybe not. For reference I found the answer to be 11/30 – manav May 21 at 12:22
• You can simply count cycles, which is 2-3 lines, at least for small numbers. There also does exist a formula for these, utilizing factorials are inclusion exclusion principle. – Quintec May 21 at 13:30
• Could somebody tell me what is a good forum to post this on? I was sent here from 'quantitative finance' page. And now it appears this too is forbidden! – manav May 21 at 14:32
• This would probably fit on math.stackexchange.com since it is a problem in discrete mathematics. – DenverCoder1 May 21 at 14:45