As a quality control procedure, a chemical lab includes in each sample to be tested a randomly chosen blank. Each blank consists of a single known chemical or a combination of two known chemicals. If protocol requires the lab to have available 12 different blanks, what is the minimum number of chemicals required to prepare the blanks?

To determine the minimum number of chemicals required to prepare 12 different blanks, we can use the concept of combinations. Let's denote the number of different chemicals as \( n \). The blanks can consist of either a single known chemical or a combination of two known chemicals. 1. **Single Chemical Blanks**: If we have \( n \) chemicals, we can create \( n \) different blanks using just one chemical. 2. **Combination of Two Chemicals**: The number of ways to choose 2 chemicals from \( n \) chemicals is given by the combination formula: \[ C(n, 2) = \frac{n(n-1)}{2} \] 3. **Total Blanks**: The total number of different blanks we can create is the sum of the single chemical blanks and the combination of two chemicals: \[ \text{Total Blanks} = n + C(n, 2) = n + \frac{n(n-1)}{2} \] 4. **Setting Up the Equation**: We need this total to be at least 12: \[ n + \frac{n(n-1)}{2} \geq 12 \] 5. **Multiplying through by 2 to eliminate the fraction**: \[ 2n + n(n-1) \geq 24 \] \[ n^2 + n - 24 \geq 0 \] 6. **Factoring the Quadratic**: \[ (n - 4)(n + 6) \geq 0 \] 7. **Finding the Roots**: The roots of the equation are \( n = 4 \) and \( n = -6 \). Since \( n \) must be a positive integer, we consider \( n \geq 4 \). 8. **Testing \( n = 4 \)**: - Single chemical blanks: \( 4 \) - Combination of two chemicals: \[ C(4, 2) = \frac{4 \cdot 3}{2} = 6 \] - Total: \( 4 + 6 = 10 \) (not enough) 9. **Testing \( n = 5 \)**: - Single chemical blanks: \( 5 \) - Combination of two chemicals: \[ C(5, 2) = \frac{5 \cdot 4}{2} = 10 \] - Total: \( 5 + 10 = 15 \) (sufficient) Thus, the minimum number of chemicals required to prepare at least 12 different blanks is \( n = 5 \).