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Combinations are a way to count how many ways you can choose items from a group, without worrying about the order of those items. For example, if you have 12 different soda cans and want to choose 3, you use combinations. Here, the combination formula is \(\binom{n}{k} \). This means you are choosing k items from a set of n items.

Mathematically, the combination formula is written as follows: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

In this problem, we have \(n = 12\) and \(k = 3\). So, we calculate \(\binom{12}{3} = \frac{12!}{3!(12-3)!} = 220\). This tells us that there are 220 ways to choose any 3 cans out of 12.

Remember: \(!\) stands for factorial, which means multiplying a series of descending natural numbers. For example, \(3! = 3 \times 2 \times 1\).

The binomial coefficient is another name for the combination formula. It plays a key role in probability and combinatorics.

In our soda can problem, we utilize the binomial coefficient to determine the number of ways to choose a subset of diet sodas and regular sodas.

For instance, to find the number of ways to choose 2 diet sodas from 3, we calculate \( \binom{3}{2} = 3\). Similarly, to choose 1 regular soda from 9, we use \( \binom{9}{1} = 9\). Multiply these numbers together to get the total combinations: \(3 \times 9 = 27\).

This shows that there are 27 different ways to select exactly 2 diet sodas and 1 regular soda from a 12-pack.

These calculations are crucial for determining probabilities because they help identify the number of favorable outcomes.

Probability is the measure of how likely an event is to occur. In our scenario, we need to find the probabilities of selecting cans under different conditions: exactly two diet sodas, exactly one diet soda, and all three diet sodas.

For each scenario, the probability is calculated by dividing the number of favorable outcomes by the total number of outcomes.

First, we calculated the total number of ways to select any three cans from the 12-pack: \( \binom{12}{3} = 220\).

Next, we calculated favorable outcomes:

• For exactly two diet sodas: \(3 \times 9 = 27\) favorable outcomes. \(\text{Probability} = \frac{27}{220} \approx 0.1227\)
• For exactly one diet soda: \(3 \times 36 = 108\) favorable outcomes. \(\text{Probability} = \frac{108}{220} \approx 0.4909\)
• For all three diet sodas: \(1 \) favorable outcomes. \(\text{Probability} = \frac{1}{220} \approx 0.0045\)

These simple calculations help us understand the likelihood of different events happening, helping solve the given problem.