Romeo and Juliet write love-letters to none but to each other. In a given period of time, Romeo writes 4 letters and Juliet writes 2 letter. During this period, at any given point of time Romeo writes greater than or equal to the number of letter written by Juliet. Find the number of ways of writing love-letters to each other.

Options

A. 10
B. 8
C. 6
D. 9

Answer

To solve the problem of how many ways Romeo and Juliet can write their love letters while ensuring that at any given point in time, Romeo has written greater than or equal to the number of letters written by Juliet, we can break it down into a systematic approach.

### Step-by-Step Solution:

1. **Identify the Letters**:
- Romeo writes 4 letters (R).
- Juliet writes 2 letters (J).
- We need to arrange these letters such that at any point in time, the number of R's is greater than or equal to the number of J's.

2. **Total Letters**:
- The total sequence of letters consists of 4 R's and 2 J's, which gives us a total of 6 letters.

3. **Case 1: Both J's are Consecutive**:
- If both J's are consecutive, we can treat the two J's as a single entity (JJ).
- Now we have 5 entities to arrange: R, R, R, R, and JJ.
- The arrangement can be visualized as follows: _ R R R R _ (with JJ being placed in the gaps).
- There are 4 gaps (before the first R, between R's, and after the last R) to place the JJ.
- We can place the JJ in any of the 3 gaps (since placing it in the first gap would violate the condition).
- Thus, the number of ways to choose 1 gap from 3 is given by \( \binom{3}{1} = 3 \).

4. **Case 2: J's are Not Consecutive**:
- If the J's are not consecutive, we can place the J's in the gaps created by the R's.
- The arrangement of R's creates 5 gaps (before the first R, between R's, and after the last R).
- We need to choose 2 out of these 5 gaps to place the J's.
- The number of ways to choose 2 gaps from 5 is given by \( \binom{5}{2} = 10 \).

5. **Total Combinations**:
- From Case 1, we have 3 arrangements.
- From Case 2, we have 10 arrangements.
- Therefore, the total number of ways to arrange the letters while satisfying the condition is \( 3 + 10 = 13 \).

6. **Final Answer**:
- The total number of ways Romeo and Juliet can write their love letters is **13**.