Question

FizzFizz soda comes in two varieties, regular and diet. If a researcher has 4 boxes of each, how many ways can he select 2 boxes of each for a quality control test?

Answer 1

The number of ways the researcher can select 2 boxes of each variety from 4 boxes of each is 36 ways, using the combination formula C(n, k) = n! / (k!(n-k)!).

Step-by-step explanation:

The question asks us to find the number of ways the researcher can select 2 boxes of regular FizzFizz soda and 2 boxes of diet FizzFizz soda from the 4 boxes of each that he has. This is a problem of combinations where the order of selection does not matter.

To determine the number of ways to select 2 boxes out of 4 for the regular variety, we use the combination formula which is C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and ! denotes factorial.

So for the regular variety: C(4, 2) = 4! / (2!(4-2)!), which simplifies to 6 ways.

We do the same calculation for the diet variety. Since the selections for regular and diet varieties are independent, we multiply the two values together to find the total number of ways to select 2 boxes of each type for quality control.

So the total number of ways is 6 (for regular) × 6 (for diet) = 36 ways.

Answer 2

Let {A, B, C, D} be the set of boxes of regular sodas and {a, b, c, d} be the set of boxes of diet sodas. Denote
`_MC_P` the numbers of combinations of M boxes taken P boxes. Note that the specific combination, for example, AB from the set of boxes of regular sodas is the same as the combination BA. The number of ways to pick 2 boxes from each category is
`_4C_2`. Hence, the number of ways of picking 4 boxes in which he pick 2 boxes from each category is
`_4C_2` *
`_4C_2` = 6*6 = 36.