Given:
The word is PROMISE.

To find:
The total number of arrangements where no two vowels come together.

Solution:
To solve this problem, we can use the concept of permutations. We will first calculate the total number of arrangements of the word PROMISE, and then subtract the number of arrangements where two vowels come together.

Step 1: Calculate the total number of arrangements
The word PROMISE has 7 letters. The total number of arrangements of these 7 letters can be calculated using the formula for permutations of n objects taken all at a time, which is n!.

Therefore, the total number of arrangements of the word PROMISE is 7!.

Step 2: Calculate the number of arrangements with two vowels together
To calculate the number of arrangements where two vowels come together, we can consider the two vowels O and I as one unit. So, we have 6 units to arrange which are P, R, M, S, (OI), and E.

The number of arrangements of these 6 units can be calculated using the formula for permutations of n objects taken all at a time, which is n!.

Therefore, the number of arrangements with two vowels together is 6!.

Step 3: Calculate the number of arrangements where no two vowels come together
To find the number of arrangements where no two vowels come together, we need to subtract the number of arrangements with two vowels together from the total number of arrangements.

Number of arrangements without two vowels together = Total number of arrangements - Number of arrangements with two vowels together

Number of arrangements without two vowels together = 7! - 6!

Step 4: Calculate the value
Now, let's calculate the value of 7! and 6!:

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Number of arrangements without two vowels together = 5040 - 720 = 4320

Therefore, the total number of arrangements where no two vowels come together is 4320.

Hence, the correct answer is option (B) 1440.