Key Concepts and Formulas

• Combinations ($^nC_r$): The number of ways to choose $r$ items from a set of $n$ distinct items without regard to order: $^nC_r = \frac{n!}{r!(n-r)!}$
• Permutations ($n!$): The number of ways to arrange $n$ distinct items in a sequence: $n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1$
• Multiplication Principle: If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.

Step 1: Analyze the word "UNIVERSE" and identify vowels and consonants

We need to determine the number of distinct vowels and consonants in the word "UNIVERSE". This is important for using combinations later.

• Distinct letters in UNIVERSE: U, N, I, V, E, R, S (7 letters)
• Vowels: U, I, E (3 vowels)
• Consonants: N, V, R, S (4 consonants)

Step 2: Select 2 vowels from the available vowels

We need to choose 2 vowels out of the 3 available. The order in which we select the vowels doesn't matter, so we use combinations.

• Number of ways to select 2 vowels from 3: $^3C_2 = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3 \times 2 \times 1}{(2 \times 1)(1)} = 3$

Step 3: Select 2 consonants from the available consonants

We need to choose 2 consonants out of the 4 available. The order in which we select the consonants doesn't matter, so we use combinations.

• Number of ways to select 2 consonants from 4: $^4C_2 = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1)(2 \times 1)} = 6$

Step 4: Arrange the 4 selected letters (2 vowels and 2 consonants)

We have selected 2 vowels and 2 consonants, giving us a total of 4 distinct letters. Now, we need to arrange these 4 letters to form a 4-letter word. The order matters, so we use permutations.

• Number of ways to arrange 4 distinct letters: $4! = 4 \times 3 \times 2 \times 1 = 24$

Step 5: Calculate the total number of 4-letter words

To find the total number of 4-letter words, we multiply the number of ways to select the vowels, the number of ways to select the consonants, and the number of ways to arrange the selected letters. This uses the multiplication principle.

• Total number of words = (Ways to select 2 vowels) $\times$ (Ways to select 2 consonants) $\times$ (Ways to arrange 4 letters)
• Total number of words = $3 \times 6 \times 24 = 432$

Common Mistakes & Tips

• Distinguish between combinations (order doesn't matter) and permutations (order matters). Select using combinations first, and then arrange using permutations.
• Remember the "without repetition" condition. It means that each letter chosen for the word must be unique. Since UNIVERSE has unique letters, this isn't an issue here.
• Break down the problem into smaller, manageable steps. Select the vowels, select the consonants, and then arrange.

Summary

We analyzed the word "UNIVERSE", identified the vowels and consonants, selected 2 vowels and 2 consonants using combinations, and then arranged the 4 selected letters using permutations. The multiplication principle was then applied to find the total number of possible 4-letter words.

The final answer is $\boxed{432}$.