Select The Correct Answer.There Are 12 Face Cards In A Standard Deck Of 52 Cards. How Many Ways Can You Arrange A Standard Deck Of 52 Cards Such That The First Card Is A Face Card?A. 12 P 1 {}_{12} P_1 12 P 1 B. 12 C 1 {}_{12} C_1 12 C 1 C. ${}_{12} P_1
Introduction
In a standard deck of 52 cards, there are 12 face cards, which include kings, queens, and jacks. When arranging the cards in a specific order, we need to consider the number of ways we can select and arrange the cards. In this article, we will explore the concept of permutations and combinations to determine the number of ways to arrange a standard deck of 52 cards such that the first card is a face card.
Understanding Permutations and Combinations
Before we dive into the problem, let's briefly review the concepts of permutations and combinations.
- Permutations: A permutation is an arrangement of objects in a specific order. The number of permutations of n objects is denoted by n! (n factorial), which is the product of all positive integers from 1 to n.
- Combinations: A combination is a selection of objects without regard to order. The number of combinations of n objects taken r at a time is denoted by C(n, r) or "n choose r," which is calculated using the formula: C(n, r) = n! / (r! * (n-r)!)
Calculating the Number of Ways to Arrange a Standard Deck of 52 Cards
Now, let's focus on the problem at hand. We want to find the number of ways to arrange a standard deck of 52 cards such that the first card is a face card.
Step 1: Selecting the First Face Card
There are 12 face cards in a standard deck of 52 cards. We need to select one of these face cards to be the first card. This can be done in 12 ways, as there are 12 face cards to choose from.
Step 2: Arranging the Remaining 51 Cards
Once we have selected the first face card, we are left with 51 cards to arrange. Since the order of the remaining cards matters, we need to calculate the number of permutations of these 51 cards.
The number of permutations of 51 objects is denoted by 51!. However, we need to consider that the first card is already fixed, so we don't need to include it in the permutation calculation.
Step 3: Calculating the Total Number of Arrangements
To find the total number of arrangements, we need to multiply the number of ways to select the first face card (12) by the number of permutations of the remaining 51 cards (51!).
The total number of arrangements is given by:
12 * 51! = 12 * 51 * 50 * 49 * ... * 2 * 1
This is equivalent to 12 * 51!, which is the product of 12 and 51!.
Conclusion
In conclusion, the number of ways to arrange a standard deck of 52 cards such that the first card is a face card is given by 12 * 51!. This is the product of 12 and 51!, which represents the number of ways to select the first face card and arrange the remaining 51 cards.
Answer
The correct answer is:
A.
This is because the number of ways to select the first face card is 12, and the number of permutations of the remaining 51 cards is 51!. Therefore, the total number of arrangements is 12 * 51!, which is equivalent to .
Discussion
This problem requires a deep understanding of permutations and combinations. The concept of permutations is essential in solving this problem, as we need to consider the order of the cards. The formula for permutations, n!, is used to calculate the number of permutations of the remaining 51 cards.
The correct answer, , represents the number of ways to select the first face card and arrange the remaining 51 cards. This is a fundamental concept in combinatorics and is used in various applications, such as probability theory and statistics.
References
- "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
- "Introduction to Combinatorics" by Kenneth P. Bogart
- "Permutations and Combinations" by Math Open Reference
Additional Resources
- Khan Academy: Permutations and Combinations
- MIT OpenCourseWare: Combinatorics
- Wolfram MathWorld: Permutations and Combinations
Q&A: Arranging a Standard Deck of 52 Cards with a Face Card as the First Card ====================================================================
Q: What is the main concept behind arranging a standard deck of 52 cards with a face card as the first card?
A: The main concept behind arranging a standard deck of 52 cards with a face card as the first card is to calculate the number of permutations of the remaining 51 cards after selecting the first face card.
Q: What is the formula for calculating the number of permutations of the remaining 51 cards?
A: The formula for calculating the number of permutations of the remaining 51 cards is 51!, which represents the product of all positive integers from 1 to 51.
Q: How many ways can we select the first face card from a standard deck of 52 cards?
A: There are 12 ways to select the first face card from a standard deck of 52 cards, as there are 12 face cards to choose from.
Q: What is the total number of arrangements of a standard deck of 52 cards with a face card as the first card?
A: The total number of arrangements of a standard deck of 52 cards with a face card as the first card is given by 12 * 51!, which represents the product of 12 and 51!.
Q: What is the correct answer to the problem of arranging a standard deck of 52 cards with a face card as the first card?
A: The correct answer is , which represents the number of ways to select the first face card and arrange the remaining 51 cards.
Q: What is the significance of permutations and combinations in solving this problem?
A: Permutations and combinations are essential concepts in solving this problem, as they help us calculate the number of ways to select the first face card and arrange the remaining 51 cards.
Q: What are some real-world applications of permutations and combinations?
A: Permutations and combinations have various real-world applications, such as probability theory, statistics, and coding theory.
Q: What are some resources available for learning more about permutations and combinations?
A: There are many resources available for learning more about permutations and combinations, including textbooks, online courses, and websites.
Q: What is the difference between permutations and combinations?
A: Permutations and combinations are related concepts, but they differ in the way they calculate the number of ways to select and arrange objects. Permutations consider the order of the objects, while combinations do not.
Q: Can you provide an example of a real-world scenario where permutations and combinations are used?
A: Yes, a real-world scenario where permutations and combinations are used is in the design of a lottery system. The lottery system uses permutations to calculate the number of possible combinations of numbers, and combinations to determine the number of winning tickets.
Q: What is the formula for calculating the number of combinations of n objects taken r at a time?
A: The formula for calculating the number of combinations of n objects taken r at a time is C(n, r) = n! / (r! * (n-r)!).
Q: What is the significance of the factorial notation in permutations and combinations?
A: The factorial notation is used to represent the product of all positive integers from 1 to n, and is essential in calculating the number of permutations and combinations.
Q: Can you provide a summary of the key concepts and formulas used in this article?
A: Yes, the key concepts and formulas used in this article are:
- Permutations: The number of ways to arrange objects in a specific order.
- Combinations: The number of ways to select objects without regard to order.
- Factorial notation: The product of all positive integers from 1 to n.
- Permutations formula: n!
- Combinations formula: C(n, r) = n! / (r! * (n-r)!)