In how many ways letters of the word 'MONDAY' can be written around a circle, if vowels are to be separated in any arrangement?

Answer

To solve the problem of arranging the letters of the word "MONDAY" around a circle with the condition that the vowels (O and A) are separated, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "MONDAY" consists of 6 letters:
- Vowels: O, A
- Consonants: M, N, D, Y

### Step 2: Arrange the consonants in a circle
Since we are arranging the letters in a circle, we can fix one consonant to avoid counting rotations as different arrangements. The number of consonants is 4 (M, N, D, Y). The number of ways to arrange n distinct objects in a circle is given by (n-1)!.

For the 4 consonants:
\[ \text{Ways to arrange consonants} = (4 - 1)! = 3! = 6 \]

### Step 3: Determine the gaps for vowels
When the consonants are arranged in a circle, they create gaps between them where the vowels can be placed. For 4 consonants, there are 4 gaps (one between each pair of consonants and one after the last consonant).

### Step 4: Choose gaps for the vowels
We need to choose 2 gaps from the 4 available gaps to place the vowels. The number of ways to choose 2 gaps from 4 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Thus, we have:
\[ \text{Ways to choose gaps} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 5: Arrange the vowels in the chosen gaps
The two vowels (O and A) can be arranged in the chosen gaps in different ways. The number of ways to arrange 2 vowels is:
\[ 2! = 2 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to arrange the consonants, the number of ways to choose the gaps, and the number of ways to arrange the vowels:
\[ \text{Total arrangements} = \text{Ways to arrange consonants} \times \text{Ways to choose gaps} \times \text{Ways to arrange vowels} \]
\[ \text{Total arrangements} = 6 \times 6 \times 2 = 72 \]

### Final Answer
Thus, the total number of ways the letters of the word "MONDAY" can be arranged around a circle with the vowels separated is **72**.

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