Probability of Not Placing Letters into Correct Envelopes Using Derangements

In this article, we will explore the problem of finding the probability that none of the 5 letters are placed into the correct envelopes using the concept of derangements. A derangement is a permutation of a set where none of the elements appear in their original position. Let's delve into the step-by-step solution to this problem.

Step 1: Total Arrangements

The first step involves calculating the total number of ways to arrange 5 letters in 5 envelopes. This is given by the factorial of the number of letters (or envelopes). The factorial (denoted as !) of a number n is the product of all positive integers less than or equal to n.

For 5 letters, the total number of arrangements (denoted as N) is:

N 5! 5 times; 4 times; 3 times; 2 times; 1 120

Step 2: Derangements

A derangement is the specific type of permutation where none of the elements appear in their original position. The formula for the number of derangements (denoted as !n) of n items is given by:

!n n! times; sum;_{i0}^{n} (-1)^i / i!

For 5 items (n 5), the number of derangements (!5) is calculated as follows:

!5 5! times; (-1^0 / 0!) - (-1^1 / 1!) (-1^2 / 2!) - (-1^3 / 3!) (-1^4 / 4!) - (-1^5 / 5!)

Let's break down the calculation step-by-step:

5! 120 !5 120 times; (-1^0 / 0!) - (-1^1 / 1!) (-1^2 / 2!) - (-1^3 / 3!) (-1^4 / 4!) - (-1^5 / 5!) -1^0 / 0! 1 / 1 1 -1^1 / 1! -1 / 1 -1 -1^2 / 2! 1 / 2 0.5 -1^3 / 3! -1 / 6 -0.1667 -1^4 / 4! 1 / 24 0.0417 -1^5 / 5! -1 / 120 -0.0083

Now, summing up these terms:

!5 120 times; (1 - 1 0.5 - 0.1667 0.0417 - 0.0083)

Cleaning up the terms inside the parentheses:

1 - 1 0 0 0.5 0.5 0.5 - 0.1667 0.3333 0.3333 0.0417 0.375 0.375 - 0.0083 0.3667

Multiplying by 120:

!5 120 times; 0.3667 44

Step 3: Probability Calculation

Finally, we can calculate the probability (denoted as P) that none of the letters are placed in the correct envelopes using the formula:

Pnone correct !5 / 5!

Substituting the values:

Pnone correct 44 / 120 11 / 30

Hence, the probability that the letters are not put into the correct envelopes is:

boxed{11 / 30}

Conclusion

Using the concept of derangements, we have determined the probability that none of the 5 letters are placed into the correct envelopes to be 11/30. This is a specific type of permutation where none of the elements appear in their original position. This problem is a classic example in combinatorics and is often used to illustrate the application of derangements in probability theory.