If 4 letters are put at random into 4 envelopes, what is the probability that at least one letter gets into the correct envelope?

In mathematics, the factorial of a non-negative integer is a fundamental concept. It is often represented by the symbol !, and it signifies the product of all positive integers less than or equal to that number. For example, the factorial of 4, denoted as 4!, is calculated as follows:
4! = 4 × 3 × 2 × 1 = 24.

The factorial function is crucial in various fields, including combinatorics, probability, and algebra. It essentially counts the number of ways to arrange a set of objects.

Key points about factorial include:

• 0! is defined to be 1 by convention.
• Factorials grow very fast—a notable property when dealing with large numbers.

Derangements are a specific type of permutation where none of the elements appear in their initial positions. Imagine you have 4 letters and 4 corresponding envelopes, and you want to ensure that no letter is placed in its correct envelope. This particular arrangement is what we call a derangement.

The formula to determine the number of derangements for n elements is:
\[!n = n! \times \bigg(\frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} ... + (-1)^n \frac{1}{n!}\bigg) \]
So, for 4 letters, the number of derangements, !4, can be calculated as:
\[!4 = 4! \times \bigg(\frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}\bigg) \]
Which simplifies to: \[!4 = 9 \].

Derangements are fascinating because they quantify scenarios where complete displacement occurs, often used in probability and combinatorial problems.

Permutations refer to the arrangement of objects in a specific order. The factorial function plays a crucial role in the calculation of permutations. If you have n distinct objects, the number of ways you can arrange these n objects is given by n!.

For instance, with 4 letters, the total number of arrangements (permutations) is: \[ 4! = 24 \].

Key characteristics of permutations include:

• Order matters in permutations.
• Unlike combinations, permutations consider different sequences of the same set as different.
• Used in fields like cryptography, game theory, and scheduling.

Permutations help us understand the total number of possible outcomes when the order of elements is significant.

In probability theory, favorable outcomes are the successful results that meet the criteria set by the problem. In our instance of placing 4 letters into 4 envelopes, the favorable outcomes would be those scenarios where at least one letter gets into the correct envelope.

To determine favorable outcomes, it's essential to subtract the unfavorable scenarios (derangements) from the total possible permutations. For example, in this problem, the total number of possible permutations, 4!, is 24, and the number of derangements (!4) is 9. Thus, the favorable outcomes are:
\[ 24 - 9 = 15 \]
The probability of at least one letter being in the correct envelope is calculated by dividing the number of favorable outcomes by the total possible outcomes:
\[ P = \frac{15}{24} = \frac{5}{8} \].

Key takeaways about favorable outcomes:

• They are specific to the problem's criteria.
• They help in determining the probability of desired events.
• Understanding and identifying them is crucial in probability calculations.