An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration. a. ?If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible? b. ?How many experimental runs are there that involve use of the lowest temperature and two lowest pressures? c. ?Suppose that five different experimental runs are to be made on the first day of experimentation. If the five are randomly selected from among all the possibilities, so that any group of five has the same probability of selection, what is the probability that a different catalyst is used on each run?

Answer : Step 1 : Given an experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration. a).If any particular experimental run involves the use of a single temperature, pressure, and catalyst. There are three different elements in this problem, so we are looking for the number of 3-tuples possible from the choices for the elements. This problem is with replacement because the choices for each of the elements can appear more than once. We can use the simplest version of the product rule to find the solution. n1=number of choices for temperature in the experiment = 3 n2=number of choices for pressure in the experiment = 4 n3=number of choices for catalyst in the experimen = 5 (n1 × n2 × n3) = Total Number of Runs So 3 × 4 × 5 = 60runs. Here 60 experimental runs. Therefore 60 experimental runs are possible. Step 2 : b). Now we have to find experimental runs are there that involve use of the lowest temperature and two lowest pressures. The set-up for this problem is identical to part a, but the variables representing the number of choices for the temperature and pressure elements have changed. n1=the lowest of the three temperatures used in the experiment =1 n2=the two lowest of the four pressures used in the experiment = 2 n3=the number of catalysts used in the experiment = 5 (n1 × n2 × n3) = Total Number of Runs 1 × 2 × 5 = 10Experimental Runs Hence 10 experimental runs are there that involve use of the lowest temperature and two lowest pressures.