How many ways can the letters in WILDCAT be arranged if the vowels cannot be next to

each other?

To determine the number of ways the letters in the word "WILDCAT" can be arranged such that the vowels are not next to each other, we can use the concept of permutations with restrictions.

First, let's identify the vowels in the word "WILDCAT." The vowels in this word are I, A, and E.

To solve this problem, we'll break it down into several steps:

Step 1: Count the total number of arrangements without any restrictions.

The word "WILDCAT" has 7 letters. Therefore, without any restrictions, the total number of arrangements is 7!.

Step 2: Count the number of arrangements where the vowels are next to each other.

To find the number of arrangements where the vowels (IAE) are together, we can consider them as a single entity. Thus, the group "IAE" can be arranged among themselves in 3! ways.

Within the "IAE" group, there are 3 possible ways to arrange these vowels (IAE, AIE, EAI) since they are different letters.

Step 3: Find the number of arrangements where the vowels are NOT next to each other.

To find the number of arrangements where vowels are NOT next to each other, we need to subtract the arrangements from Step 2 from the total number of arrangements in Step 1.

Total arrangements - Arrangements with vowels together = Arrangements with vowels NOT together

Therefore, the number of arrangements where the vowels are NOT next to each other is equal to 7! - 3!.

Step 4: Calculate the final answer.

Using factorials, we can determine the result:

7! - 3! = 5040 - 6 = 5034

So, there are 5034 ways to arrange the letters in the word "WILDCAT" such that the vowels are not next to each other.