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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 and popcorn in her box?

A. [tex]$\frac{1}{12}$[/tex]
B. [tex]$\frac{1}{6}$[/tex]
C. [tex]$\frac{1}{2}$[/tex]
D. [tex]$\frac{1}{3}$[/tex]



Answer :

GinnyAnswer
To solve this problem, let's break it down into several logical steps.

### Step 1: Determine Total Possible Combinations

First, we need to calculate the total number of possible combinations that can be created for the boxed lunches.

1. Bread Options:
- White Bread
- Wheat Bread

This gives us [tex]\(2\)[/tex] options.

2. Meat Options:
- Roast Beef
- Bologna

This also gives us [tex]\(2\)[/tex] options.

3. Snack Options:
- Chips
- Popcorn
- Pretzels

This gives us [tex]\(3\)[/tex] options.

4. Drink Options:
- Bottled Water
- Juice

This gives us [tex]\(2\)[/tex] options.

To find the total number of combinations, multiply the number of options for each category together:

[tex]\[ \text{Total Combinations} = 2 \text{ (Bread)} \times 2 \text{ (Meat)} \times 3 \text{ (Snack)} \times 2 \text{ (Drink)} = 24 \][/tex]

### Step 2: Determine Favorable Combinations

Next, we need to calculate the number of favorable combinations that meet the condition of having a roast beef sandwich and popcorn.

1. Condition for Sandwich (Roast Beef):
- White Bread with Roast Beef
- Wheat Bread with Roast Beef

This provides [tex]\(2\)[/tex] favorable options for the sandwich.

2. Condition for Snack (Popcorn):
- Only [tex]\(1\)[/tex] option for popcorn as a snack.

3. Drink Options:
- Bottled Water
- Juice

The drinks can be either, so there are [tex]\(2\)[/tex] options.

Therefore, for roast beef and popcorn, the number of favorable combinations is:

[tex]\[ \text{Favorable Combinations} = 2 \text{ (Sandwich)} \times 1 \text{ (Snack)} \times 2 \text{ (Drink)} = 4 \][/tex]

### Step 3: Calculate the Probability

Finally, we calculate the probability that Gretchen will randomly choose a boxed lunch with a roast beef sandwich and popcorn. Probability is given by:

[tex]\[ \text{Probability} = \frac{\text{Number of Favorable Combinations}}{\text{Total Number of Combinations}} = \frac{4}{24} = \frac{1}{6} \][/tex]

Therefore, the probability that Gretchen will get a boxed lunch with a roast beef sandwich and popcorn is [tex]\(\frac{1}{6}\)[/tex].

So, the correct answer is:

[tex]\[ \boxed{\frac{1}{6}} \][/tex]

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