Question

Find the number of ways you can arrange (a) all of the letters and (b) 2 of the letters in the given word. WATER

Short answer

The number of ways to arrange all letters of 'WATER' is 120, and the number of ways to arrange the letters 'W' and 'A' is 2.

Step-by-step solution

Consider all letters of the word

Let's start with 'WATER'. This word has 5 distinct letters. The number of ways to arrange all the letters is given by the factorial of the total letters, which is \( 5!\) (pronounced as '5 factorial').

Calculate the 5 factorial

The factorial of a number, in this case 5, is the product of all positive integers less than or equal to that number. So, \(5!\) is equal to \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

Consider 2 specific letters of the word

Considering only 2 letters, say 'W' and 'A'. Since there are only two letters, the number of ways to arrange these letters is given by the factorial of 2, which is \(2!\).

Calculate the 2 factorial

Similar to step 2, the factorial of 2, \(2!\), is equal to \(2 \times 1 = 2\).

Key concepts

Permutations

Understanding permutations is fundamental in solving problems related to the arrangement of objects. A permutation is a possible arrangement of items where the order matters. For example, when looking at the word 'WATER', one permuted arrangement could be 'RWATE'. Every possible rearrangement of the word represents a unique permutation.

Using factorial notation is a quick way to calculate the total number of permutations. If a set has 'n' distinct elements, the number of different permutations of these elements is denoted by 'n!'. This means for the word 'WATER', which contains 5 distinct letters, there are going to be exactly '5!' or 120 different arrangements. This concept can be applied to any scenario where you need to find out the different orders in which items can be arranged.

Combinatorics

The study of counting, arrangements, and combinations comes under the fascinating scope of combinatorics. It's a branch of mathematics that deals with complex counting problems and is essential in various fields like computer science, statistics, and physics. With combinatorics, we can systematically count the number of ways to combine or arrange objects.

In the example of arranging two letters from the word 'WATER', we're not just finding permutations of the entire word but focusing on a subset. This sub-branch of combinatorics is intuitive yet powerful for analyzing smaller selections from larger sets. Knowing how to manipulate smaller groups within a larger pool is essential in applying combinatorial principles effectively.

Algebraic Principles

Algebraic principles are the foundation of expressions, equations, and the exploration of mathematical relationships between quantities. Factorials are deeply rooted in algebra, especially in formulas and functions involving permutations and combinations. The factorial function, represented as 'n!', defines the product of all positive integers from 1 to 'n'.

In algebra, we frequently encounter factorials when expanding binomials, calculating series, and solving equations involving permutations or combinations. The calculating of factorial numbers, like in the case of solving for '5!' or '2!', is an algebraic operation that follows the principle of successive multiplication.

Probability

Probability theory is a significant domain of mathematics focused on the likelihood of events occurring. It is interwoven with combinatorics, especially when determining the probability of specific arrangements or selections from a set. In essence, probability can be calculated as the number of favorable outcomes divided by the total possible outcomes.

For example, if we need to calculate the probability of a specific two-letter arrangement from 'WATER' occurring, we'd assess it against all possible two-letter arrangements, which would be '2!'. Probability extends our understanding from merely counting possible ways to considering the chance of a single event occurring within the context of all potential events.