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?
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.
