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 52-card deck.

Step 1: Understanding the Problem

We are tasked with selecting 5 cards from a standard 52-card deck, and we need to determine the number of possible ways to do this.

Step 2: Identifying the Formula

To solve this problem, we recognize it as a combination problem because the order in which we select the cards does not matter. The combination formula is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Applying the Formula

We set \( n = 52 \) (total cards in a deck) and \( r = 5 \) (cards to be selected). Substitute these values into the combination formula: \[ C(52, 5) = \frac{52!}{5!(52-5)!} \].

Step 4: Simplifying the Calculation

We simplify \( \frac{52!}{5! \times 47!} \). Cancel out the \( 47! \) in the numerator and denominator, which results in:\[ \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} \].

Step 5: Performing the Calculation

Calculate the numerator: \( 52 \times 51 \times 50 \times 49 \times 48 = 311875200 \).Next, calculate the denominator: \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).Divide them to find the total number of combinations: \( \frac{311875200}{120} = 2598960 \).

Key Concepts

Probability

Probability embodies the likelihood of a particular event occurring. It ranges from 0 to 1, with 0 indicating an impossible event and 1 denoting certainty. For instance, when drawing a card from a well-shuffled standard deck, the probability of drawing an Ace is determined by dividing the number of Aces by the total number of cards. Since there are 4 Aces in a deck of 52 cards, the probability of this event is\[ \frac{4}{52} = \frac{1}{13} \]. Probability helps us understand how often an event might happen in the long run. Quick examples of everyday probability include predicting the weather or determining the odds of winning a game.

• Basic event: Drawing a single card from the deck
• Calculating an event: \( P = \frac{\text{favorable outcomes}}{\text{total possible outcomes}} \)
• Cumulative probability: Combining probabilities from multiple cards

Permutations

Permutations focus on arrangements where the order matters. If you need to select a certain sequence from a set, such as arranging three cards face-up from a deck, permutations are the way to go. This is because even swapping the order of two cards creates a new permutation. The formula for permutations, when drawing r items from n, is: \[ P(n, r) = \frac{n!}{(n-r)!} \] For example, if arranging 5 cards face-up from a 52-card deck, you'd calculate it as:

• Use \( P(52, 5) = \frac{52!}{(52-5)!} \)
• Understand each different order makes a new sequence

Permutations provide the tools to explore scenarios requiring specific arrangements, like assigning seating at a dinner party.

Combinatorics

Combinatorics is a branch of mathematics that studies counts, arrangements, and combinations within a set structure. It helps in addressing questions such as how to choose a subset from a larger set, without regard to order. ### CombinationsCombinations are a critical part of combinatorics as they evaluate how to select items where order does not matter. The formula is:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Applying this to the selection of 5 cards from a deck, where the outcome doesn't depend on order, is straightforward. Consider the exercise:

• Set \( n = 52 \) and \( r = 5 \)
• Substitute to get \( C(52, 5) = \frac{52!}{5!(52-5)!} = 2598960 \)

Combinatorics also involves exploring advanced topics like the pigeonhole principle and graph theory but starts here with foundational counting methods, useful for many counting problems involving different scenarios.