In how many ways we can arrange 3 red flowers, 4 yellow flowers and 5 white flowers in a row? In how many ways this is possible if the white flowers are to be separated in any arrangement ? (Flowers of same colour are identical)

Answer

To solve the problem, we will break it down into two parts:

1. **Finding the total arrangements of the flowers without any restrictions.**
2. **Finding the arrangements of the flowers with the condition that the white flowers must be separated.**

### Part 1: Total Arrangements Without Restrictions

1. **Count the total number of flowers.**
- We have 3 red flowers, 4 yellow flowers, and 5 white flowers.
- Total flowers = 3 + 4 + 5 = 12 flowers.

2. **Use the formula for arrangements of identical objects.**
- The formula for arranging n objects where there are groups of identical objects is given by:
\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3!} \]
- Here, \( n = 12 \), \( n_1 = 3 \) (red), \( n_2 = 4 \) (yellow), and \( n_3 = 5 \) (white).
- Thus, the total arrangements will be:
\[ \text{Total arrangements} = \frac{12!}{3! \times 4! \times 5!} \]

3. **Calculate the factorials.**
- \( 12! = 479001600 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 5! = 120 \)

4. **Substitute the values into the formula.**
\[ \text{Total arrangements} = \frac{479001600}{6 \times 24 \times 120} \]
\[ = \frac{479001600}{17280} = 27720 \]

### Part 2: Arrangements With White Flowers Separated

1. **Arrange the red and yellow flowers first.**
- We have 3 red and 4 yellow flowers.
- Total arrangements of red and yellow flowers:
\[ \text{Arrangements of red and yellow} = \frac{7!}{3! \times 4!} \]
- Here, \( 7 = 3 + 4 \).

2. **Calculate the arrangements.**
- \( 7! = 5040 \)
- Thus,
\[ \text{Arrangements of red and yellow} = \frac{5040}{6 \times 24} = \frac{5040}{144} = 35 \]

3. **Identify the positions for the white flowers.**
- When the red and yellow flowers are arranged, they create gaps for the white flowers.
- For example, if we arrange RRYRRY, the gaps are represented as follows:
- _ R _ R _ Y _ Y _ Y _
- There are 8 gaps (before the first flower, between flowers, and after the last flower).

4. **Choose positions for the white flowers.**
- We need to choose 5 out of these 8 gaps to place the white flowers.
- The number of ways to choose 5 gaps from 8 is given by:
\[ \text{Ways to choose gaps} = \binom{8}{5} = \binom{8}{3} = 56 \]

5. **Combine the arrangements.**
- The total arrangements with white flowers separated is:
\[ \text{Total arrangements} = \text{Arrangements of red and yellow} \times \text{Ways to choose gaps} \]
\[ = 35 \times 56 = 1960 \]

### Final Answers
- Total arrangements without restrictions: **27720**
- Total arrangements with white flowers separated: **1960**