The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants,

Q: Wait, why do we ignore the second 'E' in UNIVERSE? Shouldn't it be included?

That is a fantastic question! The key phrase here is 'without repetition'. This constraint implies that every letter in our 4-letter word must be unique. If we treated the two 'E's as distinct, we would be violating the rule that all letters must be distinct. Therefore, we only consider the set of distinct vowels {U, I, E} to ensure that any selection we make consists of unique letters.

Q: Why do we multiply by $4!$ at the end? Isn't the selection enough?

Remember, the problem asks for the number of 4-letter *words*. A word is defined by the order of its letters. If you just select the letters, you have a 'hand' of letters, but not a 'word'. Multiplying by $4!$ (which is $4 \times 3 \times 2 \times 1$) accounts for all the different ways those 4 selected letters can be arranged to form distinct words. Selection gives you the ingredients; arrangement gives you the final product!

Q: Can we just use permutations directly, like $P(n, r)$, to solve this?

You *could* try, but it gets messy because you have two different pools (vowels and consonants) with different constraints. The 'Selection-then-Arrangement' method is much safer and less prone to errors. By separating the process into two phases—first picking the letters, then arranging them—you keep the logic clean and avoid double-counting or missing combinations.

Question

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is _____.

Sigma Percentile · Accepted Answer

### The Art of Counting: Unlocking the Mystery of UNIVERSE

Combinatorics is often seen as the most intimidating branch of mathematics, but it is not just about formulas; it is about the art of counting possibilities in a structured, logical way. Today, we are going to tackle a classic problem: finding the number of 4-letter words that can be formed from the word UNIVERSE, with the constraints of having exactly 2 vowels and 2 consonants, and absolutely no repetition of letters.

### Phase 1: The Anatomy of the Word

First, let's analyze the "DNA" of the word UNIVERSE. It contains 8 letters in total, which we must categorize into two pools: vowels and consonants.

The vowels in UNIVERSE are $U, I, E,$ and $E$. The consonants are $N, V, R,$ and $S$.

Because the problem explicitly states "without repetition," we cannot treat the two $E$s as separate entities. We must refine our pools to include only distinct letters:

*   Available vowels: $\{U, I, E\}$, giving us $n_v = 3$.
*   Available consonants: $\{N, V, R, S\}$, giving us $n_c = 4$.

### Phase 2: The Selection Strategy

Now that we have our pools, we need to select our letters. We require 2 vowels and 2 consonants. We use the combination formula $^{n}C_{r}$ to select these ingredients.

To select 2 vowels from our pool of 3:

$$^{3}C_{2} = 3$$

To select 2 consonants from our pool of 4:

$$^{4}C_{2} = \frac{4 	imes 3}{2 	imes 1} = 6$$

By the Fundamental Principle of Counting, since these selections are independent, we multiply them together. The total number of ways to select our 4 letters is:

$$^{3}C_{2} 	imes ^{4}C_{2} = 3 	imes 6 = 18$$

We have identified 18 unique sets of 4 letters.

### Phase 3: The Power of Arrangement

We have our 18 sets of letters, but we must now arrange them to form words. Since we have 4 distinct letters in each set, the number of ways to arrange them in 4 slots is given by $4!$ (four factorial).

$$4! = 4 	imes 3 	imes 2 	imes 1 = 24$$

Every one of our 18 sets can be shuffled in 24 different ways to create distinct words.

### The Grand Finale

To find the total number of words, we multiply the number of selection ways by the number of arrangement ways. Our master equation is:

$$	ext{Total} = (^{3}C_{2} 	imes ^{4}C_{2}) 	imes 4!$$

Substituting our calculated values:

$$	ext{Total} = 18 	imes 24 = 432$$

The total number of 4-letter words that can be formed under these constraints is **432**. Always remember to separate the "Selection" phase from the "Arrangement" phase to keep your logic clear and error-free.

Animated Solution for Mathematics - Permutations and Combinations: The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is _____.

Enter Numerical Value:

Visualized Solution

Analyzing the Word

Extracting Distinct Vowels

Extracting Consonants

Selecting Vowels

Selecting Consonants

Total Ways of Selection

Creating the Word Slots

Arranging the Letters

The Final Equation

Computing the Values

Final Answer

The Sigma Insight: Combinations and Selection

The Art of Counting

Phase 1

Phase 2

Phase 3

The Grand Finale

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