The letters of the word DRAWER are arranged in alphabeticla ordr. The number of arrangements that precede the work REWARD is

Options

A. 241
B. 242
C. 247
D. 248

Answer

To find the number of arrangements of the letters in the word "DRAWER" that precede the word "REWARD" in alphabetical order, we can follow these steps:

### Step 1: List the letters in alphabetical order
The letters of the word "DRAWER" are D, R, A, W, E, R. Arranging these letters in alphabetical order gives us:
- A, D, E, R, R, W

### Step 2: Calculate the rank of the word "REWARD"
To find the rank of "REWARD", we need to determine how many words can be formed that come before it in alphabetical order.

1. **Fix the first letter**: The first letter of "REWARD" is R. We need to consider letters that can precede R.
- **A**: If we fix A as the first letter, the remaining letters are D, E, R, R, W. The number of arrangements of these letters is given by:
\[ \frac{5!}{2!} = \frac{120}{2} = 60 \]
- **D**: If we fix D as the first letter, the remaining letters are A, E, R, R, W. The number of arrangements is:
\[ \frac{5!}{2!} = \frac{120}{2} = 60 \]

So far, we have:
\[ 60 + 60 = 120 \text{ arrangements before starting with R.} \]

2. **Fix the second letter**: Now we consider words starting with R. The second letter in "REWARD" is E.
- **R A**: If we fix R A as the first two letters, the remaining letters are D, E, R, W. The number of arrangements is:
\[ 4! = 24 \]
- **R D**: If we fix R D as the first two letters, the remaining letters are A, E, R, W. The number of arrangements is:
\[ 4! = 24 \]

So far, we have:
\[ 120 + 24 + 24 = 168 \text{ arrangements before starting with RE.} \]

3. **Fix the third letter**: Now we consider words starting with RE. The third letter in "REWARD" is W.
- **R E A**: If we fix R E A as the first three letters, the remaining letters are D, R, W. The number of arrangements is:
\[ 3! = 6 \]
- **R E D**: If we fix R E D as the first three letters, the remaining letters are A, R, W. The number of arrangements is:
\[ 3! = 6 \]

So far, we have:
\[ 168 + 6 + 6 = 180 \text{ arrangements before starting with REW.} \]

4. **Fix the fourth letter**: Now we consider words starting with REW. The fourth letter in "REWARD" is A.
- **R E W A**: If we fix R E W A as the first four letters, the remaining letters are R. The number of arrangements is:
\[ 1! = 1 \]

So far, we have:
\[ 180 + 1 = 181 \text{ arrangements before starting with REWA.} \]

5. **Fix the fifth letter**: Now we consider words starting with REWAR. The fifth letter in "REWARD" is R.
- **R E W A R**: The only letter left is D, and there are no arrangements possible with just one letter left.

### Final Calculation
Now, we sum all the arrangements before "REWARD":
\[ \text{Total arrangements before REWARD} = 181 \]

### Conclusion
The number of arrangements that precede the word "REWARD" is **181**.