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When dealing with discrete probability, the concept of "sample space" is foundational. It's the complete set of all possible outcomes of a particular experiment. In our exercise, where two coins are drawn from a cup containing two pennies, one nickel, and one dime, each possible pair of coins forms the sample space.

Since you are drawing coins without replacement, the possible outcomes change after the first draw. To list the sample space, we consider every coin being drawn in the first position and pairing it with a different remaining coin. This gives pairing options such as:

• PP (penny first, penny second)
• PN (penny first, then nickel)
• PD (penny first, then dime)
• NP (nickel first, then penny)
• ND (nickel first, then dime)
• DP (dime first, then penny)
• DN (dime first, then nickel)

Each of these pairs is an element of the sample space, representing a unique outcome of choosing two coins in a specific order.

Probability calculation involves determining the likelihood of a given outcome in our sample space. This is usually expressed as a fraction, where the number of favorable outcomes is divided by the total number of outcomes.

In this exercise, we need to calculate the probability of drawing coins that total up to 11 cents. Since a penny is worth 1 cent and a dime is worth 10 cents, only the combinations PD (penny then dime) and DP (dime then penny) will sum up to 11 cents. From our sample space:

• PD
• DP

Thus, there are 2 favorable outcomes for drawing a total of 11 cents. Given there are 7 possible outcomes in total, the probability is calculated as:\[\text{Probability} = \frac{2}{7}\]

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of elements within a set. In the context of our exercise, combinatorics helps us systematically determine the possible outcomes of drawing coins.

When addressing the problem of choosing 2 coins in succession without replacement, we apply combinatorial principles. We start with 4 coins, and as each is chosen, the set reduces by one. Therefore, the order of choice matters, creating permutations of these choices.

The key is to ensure we consider every possible arrangement through permutation, taking into account that repetition of the same type of coin (in this case, the pennies) affects the outcome frequency. Understanding these fundamentals of combinatorics ensures that we accurately calculate probabilities and fully comprehend the array of possible events in the sample space.