Problem 46 - Calculus for Biology and Medicine

Question

In how many ways can two aces and three kings be selected from a standard deck of cards if cards are drawn without replacement?

Step-by-Step Solution

Answer

There are 24 ways to select two aces and three kings from the deck.

Step 1: Understand the problem

We need to calculate the number of ways to select two aces and three kings from a standard deck of 52 cards. Remember, the deck has four aces and four kings, and cards are drawn without replacement.

Step 2: Calculate the number of ways to choose the aces

There are four aces in the deck. We need to select two aces. This can be calculated using the combination formula \( \binom{n}{r} \), which represents the number of ways to choose \( r \) items from \( n \) items without regard to the order. So, \( \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \).

Step 3: Calculate the number of ways to choose the kings

Similarly, there are also four kings in the deck and we need to choose three. Using the combination formula again, we get \( \binom{4}{3} = \frac{4}{1} = 4 \).

Step 4: Calculate the total number of ways

To get the total number of ways to choose both the two aces and the three kings, multiply the results from Steps 2 and 3. This is because each selection of aces can be paired with any selection of kings. The calculation is \( 6 \times 4 = 24 \).

Key Concepts

Combination Formula

The combination formula is a fundamental tool in combinatorics used to find how many ways we can select items from a larger set. It's handy when the order of selection does not matter. This formula is expressed as \( \binom{n}{r} \), which reads as "n choose r." Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

• The formula is given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( ! \) denotes a factorial.
• A factorial, like \( n! \), represents the product of all positive integers up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Using this formula, you can calculate combinations easily. Let's say you need to choose 2 items from a set of 4. Plugging into the formula: \( \binom{4}{2} = \frac{4!}{2!\times(4-2)!} = \frac{24}{2 \times 2} = 6 \). This tells us there are 6 possible combinations.

Probability

Probability measures how likely an event is to occur. It's the ratio of favorable outcomes to the total possible outcomes. Expressed as a number from 0 to 1, where 0 is impossible and 1 is certain.

• The probability formula is \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
• For our card drawing problem, knowing how to calculate the combinations is key to determining likely outcomes.

When you draw cards from a deck, understanding probability helps you expect how often you'd pull a certain hand. If you're drawing without replacement, each card you draw changes the probabilities for the subsequent cards.

Deck of Cards

A standard deck of playing cards is a common tool used in combinatorial problems because it has a fixed structure, making it ideal for calculations.

• A standard deck consists of 52 cards, divided into 4 suits—hearts, diamonds, clubs, and spades.
• Each suit contains 13 cards, including numbered cards from 2 through 10, and face cards: king, queen, jack, plus the ace.

Knowing the composition of the deck is essential when calculating probabilities or combinations involving cards. Let's say you're tasked to find out how many ways you can draw two aces; you recognize there are four aces in a deck, and you can use the combination formula to discover the possible selections. Similarly, for other card drawing events, paying attention to the deck's structure gives you a strategic advantage in figuring out possible outcomes accurately.