Help!! A day later, I still do not have a solution
This is the Wordle problem from yesterday. I tried my usual first word. And hit jackpot with the second word. Got all the five letters but none of them were in the right position.
Then I got distracted. Instead of solving the Wordle problem, my idiotic brain converted it into a math problem. The question I had for myself was – in how many ways can I arrange the five letters (there is no repeat) where none of the letters will occupy the same position as the all yellow row. In other words, how many combinations of LATER can you come up with where the first letter is not L and second letter is not A and …. (you get the drift).
It is MUCH harder than I thought. One hour later, I realized I was getting late for office. Quickly solved the Wordle part and wrote to my nephew in UC Riverside before getting ready for work.
We still do not have an elegant solution. In desperation, I enumerated all possibilities. I have the answer now but do not know how to solve it.
HELP!!
1st Step:
Universal set: # of permutations with the 5 letters of ‘LATER’ = 5! = 120
2nd Step:
Of this, # of permutations with L as 1st letter = 4! = 24
For each of the five letters, this amounts to = 5 x 4! = 120
Unfortunately, when going through each letter (i.e. 2nd Step), the same word – LATER – is getting deducted multiple times. So instead of subtracting only once from the Universal set, it is getting subtracted 5 times. Hence, the answer = 5! – 5.4! + (5-1) = 4
So, the answer is 4.
Saurabh, actually, there are many more than. Here are 5 that immediately we can think of…
RLEAT
RLETA
RLATE
ELRTA
ELART
I learnt something called “derangement”! https://en.wikipedia.org/wiki/Derangement
This is fascinating! The answer is 44 as the article points out
-Rajib