Find the number of different 4-letter words (may be meaningless) that can be formed from the letters of the word 'NUMBERS'.

Answer

To find the number of different 4-letter words that can be formed from the letters of the word 'NUMBERS', we can follow these steps:

### Step 1: Identify the total number of distinct letters
The word 'NUMBERS' consists of the letters: N, U, M, B, E, R, S. There are a total of 7 distinct letters.

### Step 2: Determine the number of letters to choose
We need to form 4-letter words. Thus, we will choose 4 letters from the 7 distinct letters.

### Step 3: Use the permutation formula
Since the order of the letters matters (as we are forming words), we will use the permutation formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
where \( n \) is the total number of distinct letters, and \( r \) is the number of letters to choose.

### Step 4: Substitute the values into the formula
Here, \( n = 7 \) and \( r = 4 \). Therefore, we need to calculate:
\[ P(7, 4) = \frac{7!}{(7 - 4)!} = \frac{7!}{3!} \]

### Step 5: Calculate the factorials
Now we calculate \( 7! \) and \( 3! \):
- \( 7! = 7 \times 6 \times 5 \times 4 \times 3! \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

### Step 6: Simplify the expression
Substituting back, we have:
\[ P(7, 4) = \frac{7 \times 6 \times 5 \times 4 \times 3!}{3!} = 7 \times 6 \times 5 \times 4 \]

### Step 7: Perform the multiplication
Now we calculate:
\[ 7 \times 6 = 42 \]
\[ 42 \times 5 = 210 \]
\[ 210 \times 4 = 840 \]

### Final Answer
Thus, the total number of different 4-letter words that can be formed from the letters of the word 'NUMBERS' is **840**.

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