Question

ABC Paints Ltd. is planning to create different combination of dyes. The research team has decided they will be using five different green dyes, three different red dyes and four different blue dyes. How many combinations of dyes can be created by ABC Paints Ltd., by including at least one blue and one green dye?

• A. 5720
• B. None of the option is correct
• C. 60
• D. 31

Answer 1

The Correct Option is B

To determine the number of combinations of dyes that can be created by including at least one blue and one green dye, we can use the principle of combinatorics. Here's a step-by-step breakdown:
1. Total Dyes Available:
- 5 different green dyes
  - 3 different red dyes
  - 4 different blue dyes
2. Total Number of Ways to Choose Any Dye:
  - Each dye can either be included or not included in a combination.
  - For green dyes, there are \(2^5\) ways (since each of the 5 green dyes can either be included or not included).
  - For red dyes, there are \(2^3\) ways.
  - For blue dyes, there are \(2^4\) ways.
3. Total Number of Combinations Without Any Restrictions:
  - The total number of combinations is the product of the individual choices: \(2^5 \times 2^3 \times 2^4 = 32 \times 8 \times 16 = 4096\).
4. Subtracting Combinations That Do Not Meet the Criteria:
  - We need to exclude combinations that do not include at least one blue dye and one green dye.
  - First, calculate combinations that exclude all blue dyes: \(2^5 \times 2^3 \times 1 = 256\).
  - Second, calculate combinations that exclude all green dyes: \(1 \times 2^3 \times 2^4 = 128\).
  - Third, calculate combinations that exclude both blue and green dyes: \(1 \times 2^3 \times 1 = 8\).
5. Applying the Principle of Inclusion-Exclusion:
  - To avoid double-counting combinations that exclude both blue and green dyes, we use the inclusion-exclusion principle:
  - Total combinations excluding at least one blue or green dye: \(256 + 128 - 8 = 376\).
6. Combinations That Include at Least One Blue and One Green Dye:
  - Subtract the excluded combinations from the total combinations: \(4096 - 376 = 3720\).
Hence, the correct answer is 372, but since none of the options match this value, the correct choice is:
Answer: B) None of the options is correct.

Answer 2

Consider the dyes by colour separately. A selection can include any number of red dyes (including none), but must include at least one green dye and at least one blue dye.

Choices for red dyes (3 available, any number including none): \(2^{3}=8\) ways.
Choices for green dyes with at least one included (5 available): \(2^{5}-1=31\) ways.
Choices for blue dyes with at least one included (4 available): \(2^{4}-1=15\) ways.

Multiplying the independent choices:
\[8 \times 31 \times 15 = 3720\]

1. Option A (5720): Does not match 3720.
2. Option B (None of the option is correct): Since 3720 is not listed among the numerical options, this is the correct choice.
3. Option C (60): Does not match 3720.
4. Option D (31): This is only the count of valid green-dye choices, not the full answer, so it does not match.

The total number of valid combinations is 3720, a value that does not appear among the numerical choices.

Hence, the correct answer is Option B: None of the option is correct.