Sarvesh wants to prepare a bouquet with roses, lilies, orchids and carnations, so he walked inyo a large garden of flowers. In how many ways can he prepare a bouquet of 15 flowers such that he selects at least 1 rose, 2 lilies, 3 orchids and 4 carnations for his bouquet?

Options

A. A) 24
B. B) 120
C. C) 56
D. D) 112

Answer

To solve the problem of how many ways Sarvesh can prepare a bouquet of 15 flowers with the given constraints, we can break down the solution step by step.

### Step-by-Step Solution:

1. **Identify the Minimum Requirements**:
Sarvesh must select at least:
- 1 rose
- 2 lilies
- 3 orchids
- 4 carnations

Adding these minimum requirements gives us:
\[ 1 + 2 + 3 + 4 = 10 \text{ flowers} \]

2. **Calculate Remaining Flowers**:
Since Sarvesh wants a total of 15 flowers, we can find the remaining number of flowers he can choose after fulfilling the minimum requirements:
\[ 15 - 10 = 5 \text{ flowers} \]

3. **Define the Variables**:
Let:
- \( r \) = number of additional roses (beyond the 1 required)
- \( l \) = number of additional lilies (beyond the 2 required)
- \( o \) = number of additional orchids (beyond the 3 required)
- \( c \) = number of additional carnations (beyond the 4 required)

The equation representing the total number of additional flowers can be expressed as:
\[ r + l + o + c = 5 \]

4. **Use the Stars and Bars Theorem**:
To find the number of non-negative integer solutions to the equation \( r + l + o + c = 5 \), we can use the stars and bars combinatorial method. The formula for the number of solutions is given by:
\[ \binom{n + k - 1}{k - 1} \]
where \( n \) is the total number of items to distribute (5 flowers), and \( k \) is the number of categories (4 types of flowers).

Here, \( n = 5 \) and \( k = 4 \):
\[ \text{Number of ways} = \binom{5 + 4 - 1}{4 - 1} = \binom{8}{3} \]

5. **Calculate the Combination**:
Now, we calculate \( \binom{8}{3} \):
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

6. **Conclusion**:
Therefore, the total number of ways Sarvesh can prepare his bouquet of 15 flowers, while satisfying the given conditions, is:
\[ \boxed{56} \]