Question

How many permutations of two letters each can be formed from the letters \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e} ?\) Actually write these permutations.

Short answer

There are 20 permutations of two letters each, which are: ab, ac, ad, ae, ba, bc, bd, be, ca, cb, cd, ce, da, db, dc, de, ea, eb, ec, ed.

Step-by-step solution

Identify the quantities

Here, the total number of letters (\(n\)) is 5 (i.e. \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\)), and the number of letters to choose (\(r\)) is 2.

Apply the permutation formula

Substituting the given values in the permutation formula, we get: \[P(5,2) = \frac{5!}{(5-2)!}\]

Simplify the mathematical operation

To evaluate this, first calculate the factorial of 5 (5!), which is \(5 \times 4 \times 3 \times 2 \times 1 = 120\), and the factorial of 3 (since \(5 - 2 = 3\)), which is \(3 \times 2 \times 1 = 6\). Then divide the two results: \[P(5,2) = \frac{120}{6} = 20\]

List out the permutations

So, there should be 20 different permutations. These include: ab, ac, ad, ae, ba, bc, bd, be, ca, cb, cd, ce, da, db, dc, de, ea, eb, ec, ed.

Key concepts

Factorial

In mathematics, the factorial of a number is the product of all positive integers up to that number. Factorials are fundamental in permutations and combinations calculations. They are denoted by an exclamation mark (!). For example, the factorial of 5 is written as 5! and calculated like this:

• Start with 5 and multiply by 4.
• Continue multiplying by 3, 2, and finally 1.

This results in 5! = 120. Factorials grow very fast with increasing numbers, which is why they are so useful in counting and arranging objects. Being familiar with the factorial concept helps in making sense of more complex mathematical ideas like permutations.

Permutation Formula

The permutation formula is used to find the number of ways to arrange a subset of items from a larger set. The formula is expressed as follows: \[ P(n, r) = \frac{n!}{(n-r)!} \]Here, \(n\) is the total number of items to choose from, and \(r\) is the number of items to arrange. This formula works by calculating the factorial of the total number of items and divides it by the factorial of the difference between the total number and the number of items being arranged.
For instance, when you want to find the number of ways to arrange 2 letters out of 5, the calculation involves finding \(5!\) and \((5-2)!\). The beauty of the permutation formula is that it tells us not just about choosing items, but also concerning the order of arrangement.

Letters

Letters are often used as tangible examples to apply concepts like permutations in mathematics. They help visualize the different ways objects can be arranged or organized. Let’s take the letters \(\{a, b, c, d, e\}\), which are 5 distinct items. When asked to find permutations of these letters, especially two at a time, it's important to consider every possible pair formation. A permutation might look like 'ab' while another could be 'ba'. This highlights the role of order; 'ab' and 'ba' are two different permutations even though they consist of the same letters. Thus, letters and their arrangements help us understand the impact of order on permutations.

Combinatorics

Combinatorics is a field of mathematics that studies counting, arrangement, and combination of objects. Permutations, being one of the key topics in combinatorics, focus specifically on the arrangement of objects where order is important. Knowing how many ways you can arrange a selection of items like letters is essentially applying the principles of combinatorics. When dealing with permutations, the permutation formula (\(P(n, r) = \frac{n!}{(n-r)!}\)) is a tool that combinatorics uses extensively to solve problems. This field is not only applicable in mathematics but is also practical in real-world scenarios, such as coding, event planning, and game theory, where understanding the arrangement and combination of elements is crucial.