A clerk has 4 different letters that need to go in 4 different envelopes. What is the probability that all 4 letters are placed in the correct envelopes.

Question

We are working with permutations and combinations along with theoretical probability. 

Answer 1

 

You can solve this problem using permutations. Imagine we've put all four envelopes in a row on the table. We're going to choose letters one by one to go in each envelope in order. This means order matters; we can use permutations to figure out how many possible orderings there are.

 

(4 permute 4) = ₄P₄ = 4! = 4×3×2×1 = 24 orderings

 

That's because I have 4 choices for the first envelope's letter, then 3 choices for the second envelope's letter, then 2 remaining choices for the third envelope's letter, then 1 for the fourth envelope's letter.

 

Only 1 of those 24 orderings will put all the letters in the correct envelopes. So, the probability is 1/24.

 

 

You can also think about this another way. Let's say I'm picking the letters for the envelopes one by one. There's a 1/4 chance I pick the correct letter first. Assuming I get that one right, there's a 1/3 chance I pick the correct letter for the second envelope. Assuming I got that one right, there's a 1/2 chance I pick the correct letter for the third envelope. And assuming I got all those right, there's a probability of 1 that I pick the correct letter for the last envelope. That's a total of:

 

(1/4)(1/3)(1/2)(1) = 1/24