Permutations Question

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Introduction

Permutations are a fundamental concept in mathematics, particularly in combinatorics and probability theory. In this article, we will delve into a specific permutations question involving a standard deck of playing cards. We will explore the concept of permutations, understand the problem, and calculate the probability of a specific card sequence.

Understanding Permutations

Permutations refer to the arrangement of objects in a specific order. In the context of a standard deck of playing cards, permutations involve the arrangement of cards in a particular sequence. A standard deck of playing cards consists of 52 cards, including four suits (hearts, diamonds, clubs, and spades) with 13 cards in each suit (Ace to King).

The Problem

Three cards are dealt from a shuffled standard deck of playing cards without replacement. We are interested in finding the probability that the three cards dealt are, in order, an ace, a face card, and a 2. A face card is a card that is neither a number card nor an ace, and it includes kings, queens, and jacks.

Calculating the Probability

To calculate the probability of the specific card sequence, we need to consider the total number of possible outcomes and the number of favorable outcomes. The total number of possible outcomes is the number of ways to choose 3 cards from a deck of 52 cards without replacement, which is given by the permutation formula:

P(n, r) = n! / (n-r)!

where n is the total number of items (52 cards), and r is the number of items being chosen (3 cards).

P(52, 3) = 52! / (52-3)! = 52! / 49! = (52 × 51 × 50 × 49!) / 49! = 52 × 51 × 50 = 132,600

The number of favorable outcomes is the number of ways to choose an ace, a face card, and a 2 in that order. There are 4 aces in the deck, 12 face cards (3 in each of the 4 suits), and 4 twos in the deck. Therefore, the number of favorable outcomes is:

4 (aces) × 12 (face cards) × 4 (twos) = 192

Probability Calculation

The probability of the specific card sequence is the number of favorable outcomes divided by the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes = 192 / 132,600 = 8 / 5525

Conclusion

In this article, we explored a permutations question involving a standard deck of playing cards. We calculated the probability of a specific card sequence, which is the probability that the three cards dealt are, in order, an ace, a face card, and a 2. The probability is 8/5525, which is approximately 0.00145 or 0.145%.

Additional Information

  • Permutations vs. Combinations: Permutations and combinations are both used to calculate the number of ways to choose items from a set. However, permutations take into account the order of the items, while combinations do not.
  • Standard Deck of Playing Cards: A standard deck of playing cards consists of 52 cards, including four suits (hearts diamonds, clubs, and spades) with 13 cards in each suit (Ace to King).
  • Face Cards: Face cards are cards that are neither a number card nor an ace, and they include kings, queens, and jacks.

Frequently Asked Questions

  • What is the probability of drawing an ace, a face card, and a 2 in that order from a standard deck of playing cards?
    • The probability is 8/5525, which is approximately 0.00145 or 0.145%.
  • How many ways can you choose 3 cards from a deck of 52 cards without replacement?
    • The number of ways is given by the permutation formula: P(52, 3) = 52! / (52-3)! = 132,600.
  • What is the difference between permutations and combinations?
    • Permutations take into account the order of the items, while combinations do not.

Introduction

In our previous article, we explored a permutations question involving a standard deck of playing cards. We calculated the probability of a specific card sequence, which is the probability that the three cards dealt are, in order, an ace, a face card, and a 2. In this article, we will provide a Q&A section to address common questions and provide additional information on permutations.

Q&A

Q: What is the probability of drawing an ace, a face card, and a 2 in that order from a standard deck of playing cards?

A: The probability is 8/5525, which is approximately 0.00145 or 0.145%.

Q: How many ways can you choose 3 cards from a deck of 52 cards without replacement?

A: The number of ways is given by the permutation formula: P(52, 3) = 52! / (52-3)! = 132,600.

Q: What is the difference between permutations and combinations?

A: Permutations take into account the order of the items, while combinations do not.

Q: Can you explain the concept of face cards?

A: Face cards are cards that are neither a number card nor an ace, and they include kings, queens, and jacks.

Q: How many face cards are there in a standard deck of playing cards?

A: There are 12 face cards in a standard deck of playing cards, 3 in each of the 4 suits.

Q: What is the probability of drawing a specific card from a standard deck of playing cards?

A: The probability of drawing a specific card from a standard deck of playing cards is 1/52, since there are 52 cards in the deck.

Q: Can you provide an example of a permutation?

A: A simple example of a permutation is the arrangement of the letters in the word "CAT". The permutation of the letters in the word "CAT" is C-A-T.

Q: How do you calculate the number of permutations of a set of items?

A: The number of permutations of a set of items can be calculated using the permutation formula: P(n, r) = n! / (n-r)!, where n is the total number of items and r is the number of items being chosen.

Q: What is the relationship between permutations and factorials?

A: Permutations are closely related to factorials. The permutation formula involves factorials, and the number of permutations of a set of items can be calculated using factorials.

Additional Information

  • Permutations in Real-Life Scenarios: Permutations have numerous applications in real-life scenarios, such as scheduling, coding, and data analysis.
  • Permutations in Computer Science: Permutations are used in computer science to solve problems related to data structures, algorithms, and software engineering.
  • Permutations in Mathematics: Permutations are a fundamental concept in mathematics, particularly in combinatorics and probability theory.

Conclusion

In this article, we provided a Q&A section to address common questions and provide additional information on permutations. We hope that this article has been helpful in understanding the concept of permutations and its applications in various fields.

Frequently Asked Questions

  • What is the probability of drawing an ace, a face card, and a 2 in that order from a standard deck of playing cards?
    • The probability is 8/5525, which is approximately 0.00145 or 0.145%.
  • How many ways can you choose 3 cards from a deck of 52 cards without replacement?
    • The number of ways is given by the permutation formula: P(52, 3) = 52! / (52-3)! = 132,600.
  • What is the difference between permutations and combinations?
    • Permutations take into account the order of the items, while combinations do not.