How many combinations can be made up of `3` hens, `4` ducks and `2` geese so that each combination has hens, ducks and geese ?

Options

A. `305`
B. `315`
C. `320`
D. `325`

Explanation

To solve the problem of how many combinations can be made up of 3 hens, 4 ducks, and 2 geese such that each combination includes at least one of each type of bird, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Number of Choices for Each Bird Type**:
- For hens, we have 3 hens (h1, h2, h3).
- For ducks, we have 4 ducks (d1, d2, d3, d4).
- For geese, we have 2 geese (g1, g2).

2. **Calculate Combinations for Each Bird Type**:
- **Hens**: The number of ways to choose at least one hen from 3 hens can be calculated as:
\[ 2^3 - 1 = 8 - 1 = 7 \]
(We subtract 1 to exclude the case where no hens are chosen.)

- **Ducks**: The number of ways to choose at least one duck from 4 ducks can be calculated as:
\[ 2^4 - 1 = 16 - 1 = 15 \]

- **Geese**: The number of ways to choose at least one goose from 2 geese can be calculated as:
\[ 2^2 - 1 = 4 - 1 = 3 \]

3. **Combine the Choices**:
- Since the choices for hens, ducks, and geese are independent, we can multiply the number of combinations for each type:
\[ \text{Total combinations} = (\text{Combinations for hens}) \times (\text{Combinations for ducks}) \times (\text{Combinations for geese}) \]
\[ = 7 \times 15 \times 3 \]

4. **Calculate the Final Result**:
- Now, we perform the multiplication:
\[ 7 \times 15 = 105 \]
\[ 105 \times 3 = 315 \]

Thus, the total number of combinations that can be made up of 3 hens, 4 ducks, and 2 geese, ensuring that each combination includes at least one of each type of bird, is **315**.