In the questions number 13 how many words are there in which A is always before O ?

Options

A. 840
B. 420
C. 720
D. none of these

Answer

To solve the problem of how many words can be formed from the letters of the word "RAINBOW" such that the letter A always comes before the letter O, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the letters**: The word "RAINBOW" consists of the letters R, A, I, N, B, O, W. This gives us a total of 7 letters.

2. **Total arrangements without restrictions**: First, calculate the total number of arrangements of the letters in "RAINBOW". Since all letters are distinct, the total arrangements can be calculated using the factorial of the number of letters:
\[ \text{Total arrangements} = 7! = 5040 \]

3. **Consider the restriction (A before O)**: In any arrangement of the letters, A can either come before O or after O. Since these two scenarios are equally likely, we can find the number of arrangements where A is before O by dividing the total arrangements by 2:
\[ \text{Arrangements where A is before O} = \frac{7!}{2} = \frac{5040}{2} = 2520 \]

4. **Final Answer**: Therefore, the total number of arrangements of the letters in "RAINBOW" where A is always before O is:
\[ \text{Answer} = 2520 \]