Given 3 different red dyes, 4 different blue dyes, and 5 different green dyes, how many combinations of dyes can be made taking atleast one green and one blue dye?

Options

A. 2300
B. 31
C. 3720
D. 3560

Answer

To solve the problem of finding the number of combinations of dyes that can be made by taking at least one green and one blue dye from the given options, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Dyes Available**:
- Red Dyes: 3 different types
- Blue Dyes: 4 different types
- Green Dyes: 5 different types

2. **Calculate Combinations for Green Dyes**:
- We need to select at least one green dye from the 5 available. The total ways to select any number of green dyes (including none) can be calculated using the formula for combinations:
\[ 2^n \]
where \( n \) is the number of items to choose from. Here, \( n = 5 \):
\[ 2^5 = 32 \]
- Since we need at least one green dye, we subtract the case where no green dye is selected:
\[ 32 - 1 = 31 \]
- Thus, there are 31 ways to select at least one green dye.

3. **Calculate Combinations for Blue Dyes**:
- Similarly, for the blue dyes, we need to select at least one from the 4 available:
\[ 2^4 = 16 \]
- Again, subtracting the case where no blue dye is selected:
\[ 16 - 1 = 15 \]
- Therefore, there are 15 ways to select at least one blue dye.

4. **Calculate Combinations for Red Dyes**:
- There are no restrictions on the selection of red dyes. Thus, we can choose any combination of the 3 red dyes:
\[ 2^3 = 8 \]
- This includes the case where no red dye is selected.

5. **Combine the Results**:
- To find the total combinations of dyes, we multiply the number of ways to select the green, blue, and red dyes:
\[ \text{Total Combinations} = (\text{Ways to select green}) \times (\text{Ways to select blue}) \times (\text{Ways to select red}) \]
\[ = 31 \times 15 \times 8 \]

6. **Calculate the Final Result**:
- First, calculate \( 31 \times 15 \):
\[ 31 \times 15 = 465 \]
- Next, multiply this result by 8:
\[ 465 \times 8 = 3720 \]

### Final Answer:
The total number of combinations of dyes that can be made by taking at least one green and one blue dye is **3720**.

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