How many words can be formed with the letters of the word 'NUMBERS', if all consonants are always together?

Answer

To solve the problem of how many words can be formed with the letters of the word 'NUMBERS' if all consonants are always together, we can follow these steps:

### Step 1: Identify the letters in 'NUMBERS'
The word 'NUMBERS' consists of 7 letters: N, U, M, B, E, R, S.

### Step 2: Separate vowels and consonants
In 'NUMBERS', the vowels are:
- U, E

The consonants are:
- N, M, B, R, S

### Step 3: Treat all consonants as a single unit
Since we want all consonants to be together, we can treat the group of consonants (N, M, B, R, S) as a single unit. This means we can represent the arrangement as:
- (NMBRS), U, E

Now we have 3 units to arrange:
1. The consonant unit (NMBRS)
2. The vowel U
3. The vowel E

### Step 4: Calculate the arrangements of the units
The number of ways to arrange these 3 units is given by:
\[ 3! = 6 \]

### Step 5: Arrange the consonants within their unit
Now, we need to arrange the consonants within their unit (N, M, B, R, S). The number of ways to arrange these 5 consonants is given by:
\[ 5! = 120 \]

### Step 6: Calculate the total arrangements
To find the total number of arrangements of the letters in 'NUMBERS' with all consonants together, we multiply the arrangements of the units by the arrangements of the consonants:
\[ \text{Total arrangements} = 3! \times 5! = 6 \times 120 = 720 \]

### Final Answer
Thus, the total number of words that can be formed with the letters of the word 'NUMBERS' if all consonants are always together is **720**.

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