The letters of the word EQUATION are arranged in a row. Find the probability that arrangements start with a vowel and end with a consonant.

Answer

To solve the problem of finding the probability that the arrangements of the letters in the word "EQUATION" start with a vowel and end with a consonant, we can follow these steps:

### Step 1: Identify the letters in the word "EQUATION"
The word "EQUATION" consists of the following letters:
- Vowels: E, U, A, I, O (5 vowels)
- Consonants: Q, T, N (3 consonants)

### Step 2: Count the total number of letters
The total number of letters in "EQUATION" is 8.

### Step 3: Calculate the total possible arrangements
The total number of arrangements of the 8 letters is given by the factorial of the number of letters:
\[ \text{Total arrangements} = 8! = 40320 \]

### Step 4: Determine the conditions for favorable outcomes
We need to find the arrangements that start with a vowel and end with a consonant.

1. **Choose the first letter (vowel)**: There are 5 options (E, U, A, I, O).
2. **Choose the last letter (consonant)**: There are 3 options (Q, T, N).
3. **Arrange the remaining letters**: After choosing the first and last letters, we have 6 letters left to arrange.

### Step 5: Calculate the number of favorable outcomes
The number of ways to select the first letter (vowel) and the last letter (consonant) and arrange the remaining letters is calculated as follows:
\[ \text{Number of favorable outcomes} = (\text{number of vowels}) \times (\text{number of consonants}) \times (\text{arrangements of remaining letters}) \]
\[ = 5 \times 3 \times 6! \]
Calculating \(6!\):
\[ 6! = 720 \]
Thus, the number of favorable outcomes becomes:
\[ = 5 \times 3 \times 720 = 10800 \]

### Step 6: Calculate the probability
The probability \(P\) that the arrangements start with a vowel and end with a consonant is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total arrangements}} = \frac{10800}{40320} \]

### Step 7: Simplify the probability
To simplify:
\[ P = \frac{10800 \div 10800}{40320 \div 10800} = \frac{1}{3.7333} \approx \frac{15}{56} \]

### Final Answer
Thus, the probability that the arrangements start with a vowel and end with a consonant is:
\[ \frac{15}{56} \]