Question

How many words can be formed from the letters of the word EQUATION using any four letters in each word? (1) 840 (2) 1680 (3) 2080 (4) 3050

Short answer

Answer: None of the given options provides the actual answer, which is 1320. However, if forced to choose among the options, option (1) 840 is the closest answer.

Step-by-step solution

Identify the Total Number of Letters

The given word, EQUATION, has a total of 8 letters. It's important to consider that E occurs twice in the word.

Calculate the Number of Words Without Repetition

First, we'll find out how many words can be formed using any four letters with no repetition of E in the word. We have 7 different letters (assuming E is a single entity) and we need to select 4 which can be calculated using combinations: C(7, 4) = 7! / (3! * 4!) = 35.

Calculate the Number of Words With Repetition

Now, let's calculate the number of words formed using any four letters with one of the E's repeated. We have a total of 6 letters (remaining letters excluding any E) to choose from and we need to choose 3 which can be calculated using combinations: C(6, 3) = 6! / (3! * 3!) = 20.

Multiply Choices by the Number of Permutations

For each combination found in steps 2 and 3, there are 4! permutations since we're forming new words with 4 letters. Therefore, in the case of no repetitions, we will have 35 * 4! and with repetitions, we will have 20 * 4!.

Add the Results Together

The total number of words that can be formed using any four letters from the word EQUATION is the sum of the results from steps 4 for the non-repeated and repeated words: Total = 35 * 4! + 20 * 4! Total = 35 * 24 + 20 * 24 Total = 840 + 480 Total = 1320 The correct option is not among the available options. However, if the available options had to be chosen, option (1) 840 would be the closest answer.

Key concepts

Permutations

Permutations are arrangements of objects in a specific order. In the context of forming words, each different arrangement of letters represents a permutation. When calculating permutations, every different sequence of a set of letters counts as a unique arrangement.
For permutations, the order matters. This means that the word "EQUA" is different from "AQEU" even though they consist of the same letters. To find the number of different permutations of a set number of items, we use the factorial notation. For example, if we have 4 letters to arrange, we calculate the permutations by 4! (which is 4 × 3 × 2 × 1), resulting in 24 different arrangements.

• Key Point: Permutations are used when the order of items is important.

Understanding permutations is essential in solving many combinatorial problems because they determine how we can arrange a selection of items.

Combinations

Combinations differ from permutations in that the order of the items does not matter. When solving the example problem where we form words from the word EQUATION, initially, we need to choose 4 letters from the available set. This is where combinations come into play.
In combinations, we are interested in the selection itself and not the order of selection. The formula for combinations is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose. So for our problem, using 7 different letters to choose 4 means \( C(7, 4) \).

• Key Point: Use combinations when the selection of items is important but not their order.

Combinations are fundamental in calculating how many subsets or groups can be formed from a larger set.

Probability

Probability in combinatorics often deals with the likelihood of forming certain groups or arrangements. However, in this problem, probability per se isn't directly calculated, but understanding it can be helpful in similar scenarios.
For instance, if you wanted to calculate the probability of picking a certain type of word from all possible 4-letter combinations from "EQUATION," knowing the count of favorable permutations becomes essential. Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

• Key Point: While probability is not calculated here, understanding it helps in determining the likelihood of particular arrangements.

In broader contexts, probability helps to understand how likely a particular combination or permutation is among all possible options.

Algebra

Algebra involves using mathematical operations and relationships to solve problems, often involving variables and symbols. In the context of our exercise, algebra is implicitly used when solving combinations and permutations through formulas.
Using factorial notation, such as \(7!\) or \(4!\), is applying algebraic principles to find the total number of arrangements or selections of letters. These calculations showcase algebra's power in breaking complex problems into simpler, systematic steps.

• Key Point: Algebraic methods are crucial for calculating combinations and permutations through systematic formulas.

This mathematical toolset empowers us to solve problems efficiently, ensuring all possible scenarios are considered systematically.