A secretary has typed five letters to five different people, an addressed an envelope for each letter. The letter is put into the letter at random.

Question

How many different assignments of letters to envelopes are possible? What is the probability that each person receives the correct letter?

Answer 1

Let's imagine a similar problem:  You are given 5 slips of paper, each with a number from 1 to 5 written on it and then folded so the number can't be seen.  On a table are five boxes, also each with a number from 1 to 5. You start by dropping a piece of paper into box 1, then into box 2, and continue until you've dropped the last piece of paper into box 5.

 

To see what's going on, we can view some possibilities.

 

Box 1      Box 2      Box 3      Box 4     Box 5

1               2              3              4              5

2               3              5              4              1

3               1              2              5              4.............

 

The numbers below the boxes represent the number on the piece of paper which you've placed into that box.

 

If we look at each row as a list, we'd have

row 1:  1,2,3,4,5

row 2:  2,3,5,4,1

row 3:  3,1,2,5,4

 

We can see that each list contains exactly the same elements, but in a different order.  The ordering is important (in fact, it's the only thing which makes each list distinct).  What we use to figure out the number of arrangements in such a situation is called a permutation, often written as

 

_nP_r, where n is the total number of elements we have to choose from, and r is the number of elements in the set we're creating (here, that would be the list). 

 

The total number of elements used in our example is 5, and we're creating lists of 5 elements.  So, we have

 

₅P₅.  The formula for a permutation is n!/(n-r)!.  Here, that comes out to 5!/(5-5)!.  Using the convention that 0! = 1, we have 5! = 5x4x3x2x1 possibilities.

 

The only possibility where the number written on the slip of paper is the same as the number on the box would correspond to the list 1,2,3,4,5.  So, there is only way for such an arrangement to occur.

 

The probability that this arrangement will occur is

 

(number of ways for the arrangement to occur)/(number of possible arrangements),

or 1/5!.

 

This is exactly the same as the probability that each person receives the correct letter in your original problem.  I believe the secretary will soon be looking for new work.