The number of words that can be formed using all the letters of the word REGULATIONS such that `G` must come after `R`, `L` must come after `A`, and `S` must come after `N` are

Options

A. `11!//8`
B. `11!`
C. `"^(11)P_(6)`
D. none of these

Explanation

To solve the problem of finding the number of words that can be formed using all the letters of the word "REGULATIONS" under the given constraints, we will follow these steps:

### Step 1: Count the total letters and identify constraints
The word "REGULATIONS" consists of 11 letters: R, E, G, U, L, A, T, I, O, N, S. The constraints given are:
- G must come after R
- L must come after A
- S must come after N

### Step 2: Calculate the total arrangements without constraints
First, we calculate the total number of arrangements of the letters in "REGULATIONS". Since all letters are unique, the total arrangements are given by the factorial of the number of letters:

\[ \text{Total arrangements} = 11! = 39916800 \]

### Step 3: Apply the constraints
Now, we need to account for the constraints. Each constraint effectively reduces the number of valid arrangements.

1. **For the constraint G after R**:
- In any arrangement of the letters, R and G can be in any order. Since there are two positions for R and G, half of the arrangements will have G after R.
- Therefore, the number of arrangements satisfying this condition is:

\[ \frac{11!}{2} = \frac{39916800}{2} = 19958400 \]

2. **For the constraint L after A**:
- Similarly, for L and A, half of the arrangements will have L after A. So we further divide by 2:

\[ \frac{19958400}{2} = 9979200 \]

3. **For the constraint S after N**:
- Again, for S and N, half of the arrangements will have S after N. We divide by 2 one more time:

\[ \frac{9979200}{2} = 4989600 \]

### Step 4: Final count of valid arrangements
Thus, the total number of arrangements of the letters in "REGULATIONS" that satisfy all the given conditions is:

\[ \text{Valid arrangements} = 4989600 \]

### Conclusion
The number of words that can be formed using all the letters of the word "REGULATIONS" such that G comes after R, L comes after A, and S comes after N is **4989600**.