Introduction:
In this problem, we need to determine the number of ways to select six letters, including at least one vowel, from the given word "INCOMPUTABLE". Let's break down the problem into smaller steps to find the solution.

Step 1: Counting the total number of letters:
To begin, we need to count the total number of letters in the word "INCOMPUTABLE". By counting, we find that there are 12 letters in total.

Step 2: Counting the number of vowels:
Next, we count the number of vowels in the word "INCOMPUTABLE". The vowels present in the word are "I", "O", "U", and "A", with a total count of 4.

Step 3: Selecting all consonants:
Since we need at least one vowel in our selection, let's first consider selecting all six consonants. The word "INCOMPUTABLE" has 8 consonants: "N", "C", "M", "P", "T", "B", "L", and "T". To select six consonants out of eight, we can use the combination formula.

Combination Formula:
The combination formula is given by C(n, r) = n! / (r!(n-r)!), where n is the total number of objects, and r is the number of objects to be selected.

Using the combination formula, we can calculate the number of ways to select six consonants out of eight:

C(8, 6) = 8! / (6!(8-6)!)
= 8! / (6!2!)
= (8 * 7 * 6!) / (6! * 2 * 1)
= (8 * 7) / (2 * 1)
= 28

Therefore, there are 28 ways to select all six consonants.

Step 4: Selecting at least one vowel:
Now, we need to calculate the number of ways to include at least one vowel in our selection. To do this, we subtract the number of ways to select all consonants from the total number of ways to select six letters.

Total number of ways to select six letters = Total number of ways to select all consonants

Therefore, the number of ways to select at least one vowel = Total number of ways to select six letters - Total number of ways to select all consonants

= C(12, 6) - C(8, 6)

Calculating C(12, 6):
Using the combination formula, we can calculate the number of ways to select six letters out of twelve:

C(12, 6) = 12! / (6!(12-6)!)
= 12! / (6! * 6!)
= (12 * 11 * 10 * 9 * 8 * 7 * 6!) / (6! * 6!)
= 12 * 11 * 10 * 9 * 8 * 7
= 665,280

Calculating C(8, 6):
We have already calculated this in Step 3, and it is equal to 28.

Final