This is a discrete math (counting techniques) problem. Please explain clearly, no cursive writing. **Additional Instructions**For the exercises that ask you to answer "How many ...", express your results as a combinatorial formula, calculate the integer value that answers the question, and provide a short justification of your formula choice. **Title: Dealing Cards from a Standard Playing Deck****Introduction**In card games, distributing cards among players is a fundamental aspect that can be explored through combinatorial mathematics. A standard playing deck consists of 52 cards. Let's explore the problem of dealing hands to four players under different conditions.**Problem Statement**How many ways are there to deal hands from a standard playing deck to four players if:1. Each player gets exactly 13 cards.2. Each player gets seven cards and the rest of the cards remain in the deck.**Discussion**1. **13 Cards per Player**: When dealing 13 cards to each player, the entire deck is distributed among the players, leaving no cards remaining. This scenario reflects a balanced distribution where each player has an equal share of the deck.2. **Seven Cards per Player**: In this case, each player receives seven cards, totaling 28 cards, with the remaining 24 cards staying in the deck. This setup may be used in games that involve holding a smaller hand or in preliminary rounds.Understanding these distributions helps in strategizing and predicting possible outcomes in card games. Analyzing the mathematical possibilities also enriches one's grasp of probability and combinatorial theory.

Name: This is a discrete math (counting techniques) problem. Please explain
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: This is a discrete math (counting techniques) problem. Please explain clearly, no cursive writing. **Additional Instructions**For the exercises that ask you to answer "How many ...", express your results as a combinatorial formula, calculate the inte

Consider the provided question, Given, 4 players are playing cards. (1) The number of ways each player gets exactly 13 cards is calculated as follows: the total number of cards in deck is 52 cards. the number of ways of combination of r elements out of n elements is Crn=n!r!×n-r!. The number of ways the 1st player get exactly 13 cards out of 52 cards is C1352=52!13!×52-13!=52!13!×39! ways The remaining cards are 52-13=39 cardsThe number of ways the 2nd player get exactly 13 cards out of 39 cards is C1339=39!13!×39-13!=39!13!×26! ways The remaining cards are 39-13=26 cards The number of ways the 3rd player get exactly 13 cards out of 26 cards is C1326=26!13!×26-13!=26!13!×13! ways The remaining cards are 26-13=13 cards The number of ways the 4th player get exactly 13 cards out of 13 cards is C1313=13!13!×13-13!=13!13!×0! ways Then, the total number of ways each player gets exactly 13 cards is 52!13!×39!×39!13!×26!×26!13!×13!×13!13!×0!=52!13!×13!×13!×13! Therefore, the total numbers of ways each player gets exactly 13 cards is 52!13!×13!×13!×13! ways. (2) The number of ways each player gets 7 cards and rest of the card remain in the deck is calculated as follows: the total number of cards in deck is 52 cards. The number of ways the 1st player get 7 cards out of 52 cards is C752=52!7!×52-7!=52!7!×45! ways The remaining cards are 52-7=45 cards The number of ways the 2nd player get 7 cards out of 45 cards is C745=45!7!×45-7!=45!7!×38! ways The remaining cards are 45-7=38 cardsThe number of ways the 3rd player get 7 cards out of 38 cards is C738=38!7!×38-7!=38!7!×31! ways The remaining cards are 38-7=31 cards The number of ways the 4th player get 7 cards out of 31 cards is C731=31!7!×31-7!=31!7!×24! ways Then, the total number of ways each player gets 7 cards is 52!7!×45!×45!7!×38!×38!7!×31!×31!7!×24!=52!7!×7!×7!×24! Therefore, the total numbers of ways each player gets 7 cards is 52!7!×7!×7!×24! ways.