In designing an experiment, the researcher can often choose many different levels of the various factors in order to try to find the best combination at which to operate. As an illustration, suppose the researcher is studying a certain chemical reaction and can choose four levels of temperature, five different pressures, and two different catalysts.

(a) To consider all possible combinations, how many experiments would need to be conducted?

(b) Often in preliminary experimentation, each factor is restricted to two levels. With the three factors noted, how many experiments would need to be run to cover all possible combinations with each of the three factors at two levels?

a) 40 possible combinations

b) 8 possible combinations

Step-by-step explanation:

Given:

- Levels of temperature = 4

- Levels of catalysts = 2

- Levels of pressure = 5

Find:

To consider all possible combinations, how many experiments would need to be conducted?

With the three factors noted, how many experiments would need to be run to cover all possible combinations with each of the three factors at two levels?

Solution:

- We experiment each of the 3 factors to a certain level. So to conduct any experiment we have:

     Temperature = 4C1 = 4 possibilities

     Catalysts = 2C1 = 2 possibilities

     Pressure = 5C1 = 5 possibilities.

- Hence, the total outcome for one particular experiment is a combination of all 3 factors possibility:

    Total possible outcomes = 4*2*5 = 40 possible combinations

- Similarly if all factors are restricted to two levels only we have:

     Temperature = 2C1 = 2 possibilities

     Catalysts = 2C1 = 2 possibilities

     Pressure = 2C1 = 2 possibilities.

- Hence, the total outcome for one particular experiment is a combination of all 3 factors possibility:

    Total possible outcomes = 2*2*2 = 8 possible combinations