Question

the cafeteria creates pre - made boxed lunches with equal numbers of the following items: a sandwich made with either white or wheat bread and either roast beef or bologna a snack that is either chips, popcorn, or pretzels a drink that is either bottled water or juice if gretchen randomly chooses one of the boxed lunches, what is the probability that she will get a roast beef sandwich on wheat bread and popcorn in her box? 1/24 3/4 1/6 1/12

Explanation:

Step1: Calculate the number of sandwich combinations

There are 2 types of bread (white or wheat) and 2 types of fillings (roast beef or bologna). The number of sandwich combinations is \(2\times2 = 4\) (using the counting principle \(m\times n\) where \(m\) is the number of bread - type choices and \(n\) is the number of filling - type choices).

Step2: Calculate the number of snack combinations

There are 3 types of snacks (chips, popcorn, or pretzels). So, the number of snack combinations is \(3\).

Step3: Calculate the number of drink combinations

There are 2 types of drinks (bottled water or juice). So, the number of drink combinations is \(2\).

Step4: Calculate the total number of box - lunch combinations

Using the counting principle (if there are \(a\) ways to do one thing, \(b\) ways to do a second thing, and \(c\) ways to do a third thing, then there are \(a\times b\times c\) ways to do all three things together), the total number of box - lunch combinations is \(4\times3\times2=\ 24\).

Step5: Calculate the number of favorable combinations

The favorable combination: 1 type of sandwich (roast - beef on wheat), 1 type of snack (popcorn), and 2 types of drinks. The number of favorable combinations is \(1\times1\times2 = 2\).

Step6: Calculate the probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So, \(P=\frac{2}{24}=\frac{1}{12}\).

Answer:

\(\frac{1}{12}\) (the fourth option)