Problem 22 - Calculus for Biology and Medicine

Question

A standard deck contains 52 different cards. In how many ways can you select 5 cards from the deck?

Step-by-Step Solution

Answer

There are 2,598,960 ways to select 5 cards from a deck of 52 cards.

Step 1: Understand the Problem

To solve this problem, we need to find the number of ways to select 5 cards from a deck of 52 cards. This scenario represents a combination where the order of selection does not matter.

Step 2: Apply Combination Formula

The formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. In this problem, \( n = 52 \) and \( r = 5 \).

Step 3: Calculate the Factorials

Compute the factorials needed for the formula: \( 52! \), \( 5! \), and \( (52-5)! = 47! \). However, for practical calculation, manual calculation of full \( 52! \) is not needed as it will be reduced in the combination formula.

Step 4: Plug Values into the Combination Formula

Substitute the values into the combination formula: \[ C(52, 5) = \frac{52!}{5! \times 47!} \]

Step 5: Simplify the Formula

Recognize that \( 52! = 52 \times 51 \times 50 \times 49 \times 48 \times 47! \), allowing \( 47! \) to cancel out in the numerator and denominator:\[ C(52, 5) = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} \]

Step 6: Calculate the Final Result

Perform the arithmetic:- \( 52 \times 51 \times 50 \times 49 \times 48 = 311875200 \)- \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \)- Divide them: \( \frac{311875200}{120} = 2598960 \)Therefore, the number of ways to select 5 cards from a deck of 52 is 2,598,960.

Key Concepts

Combination Formula

Combinations are essential when it comes to probabilities because they help us determine how many ways we can choose a subset of items from a larger set, where the order of items does not matter. This is often represented by the formula \[ C(n, r) = \frac{n!}{r!(n-r)!} \] Here, **\( n \)** is the total number of items available, and **\( r \)** is the number of items we wish to choose.

• Key Point: The combination formula calculates the number of different groups or sets that can be formed from a larger set.
• Order Doesn’t Matter: If the order was important, we would be dealing with permutations instead of combinations.
• Real Life Usage: Anytime you need to select members for a team, choose cards, or form committees where order isn't relevant, combinations are your tool.

In our exercise, we use \( n = 52 \) (cards) and \( r = 5 \) (cards to select). By calculating \( C(52, 5) \) using the formula, we determine there are 2,598,960 possible ways to choose those 5 cards. Thus, mastering this concept is crucial in set selections and many probability problems.

Factorials

Factorials are a vital mathematical operation in combinatorics and probability, often represented by an exclamation mark '!', such as \( n! \). It means multiplying together all whole numbers from 1 up to \( n \).

• Definition: \( n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1 \)
• The Simple Case: 0! is defined to be 1. This might seem odd, but it's essential for mathematical consistency.
• Usage: Factorials are used to find out how many different ways items can be arranged, and are crucial in the combination and permutation formulas.

In our combination formula, factorials help to manage these arrangements. For instance, \( 52! \) is massive and represents the arrangements of a whole deck of cards, while \( 5! \) is just for our hand. Although you don’t calculate the full \( 52! \) here, understanding its role in simplifying the formula is key.

Basic Probability

Basic probability helps us understand the likelihood of events occurring. Probability follows a simple rule: it's the ratio of the number of favorable outcomes to the total number of possible outcomes.

• Formula: Probability (\( P \)) = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)
• Between 0 and 1: Probability values range between 0 and 1, where 0 means the event is impossible, and 1 means it's certain.
• Complementary Rule: The probability that an event does not happen is \( 1 - P \).

While our exercise focuses on combinations, the concept of probability is closely tied. Once the combinations (possible outcomes) are known, probabilities can help determine chances, such as the likelihood of a specific hand in a card game. Understanding how combinations work hand-in-hand with probability offers better insights into real-world scenarios.