Question

Two cards are to be drawn in order from a pack of 4 cards (say, an ace, king, queen, and jack), the drawn card not being replaced before the second card is drawn. How many different drawings are possible?

Short answer

There are 12 different possible drawings.

Step-by-step solution

Identify the number of options for the first card drawn

For the first card drawn, all 4 cards are available, so there are 4 possibilities.

Identify the number of options for the second card drawn

As the first card was not replaced, there are only 3 cards left. Thus, for the second card, we have 3 possibilities.

Calculate the total different possibilities

We use the rule of product, which states that if there are m ways to do one thing, and n ways to do another, then there are m*n ways to do both. Here, for the first card drawn, we have 4 options and for the second, we have 3. So multiplying these together, we get \(4 * 3 = 12\) total different possible outcomes.

Key concepts

Understanding Permutation

When dealing with ordering or arranging objects, the mathematical concept of permutation comes into play. A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For instance, if we're looking at our exercise with the 4 cards, we are engaging with permutations when we draw two cards one after another without replacement. This means that the order in which we draw the cards matters, where drawing an Ace first and a King second is different from drawing a King first and an Ace second.

Additionally, permutations can be calculated using factorial notation. For our exercise, since we are not using all the cards from the set, we are dealing with a partial permutation; the general formula for which is olimits P(n, r) = \( \frac{n!}{(n-r)!} \), where n is the total number of items to choose from, and r is the number of items to arrange. In this case, we are arranging 2 cards (r) from the total of 4 (n), resulting in \( P(4, 2) = \frac{4!}{(4-2)!} = \frac{4!}{2!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1} = 4 \times 3 = 12 \) permutations.

Calculating Probability

Probability is a measure of the likelihood that an event will occur. It's calculated as the number of favorable outcomes divided by the total number of possible outcomes. In our card-drawing exercise, if we were to ask the probability of drawing any specific combination of cards, we'd look at the number of ways that specific event can occur compared to all possible events.

To visualize this, suppose we wanted to know the probability of drawing an Ace followed by a King. Since there are 12 different drawings possible (as per the permutation calculated above), and there is only 1 way to draw the Ace followed by the King, the probability would be \( \frac{1}{12} \). It's crucial to recognize that these calculations assume all outcomes are equally likely, which is true when drawing cards from a shuffled deck. Understanding probability not only helps to calculate the likelihood of events but also informs decision-making under uncertainty.

Applying the Rule of Product

The rule of product, also known as the multiplication principle, is a fundamental rule in counting that allows the calculations of the total number of outcomes for consecutive events. The rule states that if event A can occur in m ways and event B can occur in n ways, then the two events can occur in m*n ways, assuming the two events are independent.

In the card drawing exercise, the first card can be drawn in 4 ways, and for each of these ways, the second card can be drawn in 3 ways. This gives us a total of \(4 \times 3 = 12\) ways to draw two cards in order without replacement. This is a direct application of the rule of product. It is important to note that 'without replacement' indicates the two draws are not independent, but the rule of product still applies because the number of ways to perform the second task changes in a predictable manner after the first card is drawn. The rule of product is a powerful tool for solving complex problems by breaking them into simpler, sequential tasks.