To solve the problem of finding the number of ways to arrange the letters in "EQUATION" such that no two vowels are adjacent, follow these steps:

Step 1: Identify Vowels and Consonants

The word "EQUATION" consists of 8 letters:

• Vowels: E, U, A, I, O (5 vowels)
• Consonants: Q, T, N (3 consonants)

Step 2: Key Insight for Non-Adjacent Vowels

For no two vowels to be adjacent, we first arrange the consonants, then place vowels in the gaps between them (including ends).

• Number of gaps between C consonants = C + 1.
• Here, C = 3 → gaps = 3 + 1 = 4.

Step 3: Feasibility Check

Each gap can hold at most 1 vowel (to avoid adjacency). Thus, the maximum number of vowels that can be placed without adjacency is 4.

But we have 5 vowels, which exceeds the number of available gaps.

Thus, it is impossible to arrange the vowels without adjacency.

Answer: 0

$\boxed{0}$

作者声明:本文包含人工智能生成内容。