Problem 105 - Algebra for JEE ADVANCED

Question

Two packs of 52 cards are shuffled together. The number of ways in which a man can be dealt 26 cards so that he does not get two cards of the same suit and same denomination is a. \({ }^{\$ 7} C_{26} \cdot 2^{26}\) b. \({ }^{104} C_{26}\) c. \(2 \cdot{ }^{32} C_{26}\) d. none of these

Step-by-Step Solution

Answer

d. none of these

Step 1: Understand the Problem

We need to find the number of ways to deal 26 cards to a person from two shuffled decks such that no two cards are of the same suit and denomination.

Step 2: Consider Two Identical Cards

In two decks of 52 cards each, there are two of each card type available (e.g., two Ace of Spades, two Two of Clubs, etc.). We must ensure when dealing 26 cards, the person does not receive two cards of the same type.

Step 3: Total Unique Cards Available

Since each card type appears exactly twice and there are 52 unique card types, the total number of unique cards available is 52.

Step 4: Select 26 Unique Cards

To ensure no two cards of the same type are selected, we must choose 26 unique cards from the 52 unique cards. This can be done in \({}^{52} C_{26}\) ways.

Step 5: Validate the Options

Review the given options to determine if they match our solution. None of the options provided match the calculation \({}^{52} C_{26}\).

Step 6: Conclusion

Since none of the choices match the number of successful ways, the answer is option d, 'none of these.'

Key Concepts

Card Counting

Card counting is a technique often used to track the cards that have been dealt in card games. It is a strategic approach utilized by players to assess the probability of certain outcomes. In this exercise, we're essentially "counting" the different types of cards that can be dealt to ensure that no two cards of the same suit and denomination are given to one person.

When dealing with two shuffled decks, each having 52 cards, there are two of each card, such as two Ace of Spades. To effectively count the cards, it helps to first understand that:

• There are 52 unique types of cards.
• Even though there are 104 cards in total, we are interested in a unique 52 cards for counting purposes.

Card counting, therefore, becomes crucial in making sure the chosen cards (26 in this case) do not contain duplicates of the same type.

Permutations and Combinations

Permutations and combinations are two key concepts in combinatorics. They help determine the number of possible arrangements or selections of items.

In our exercise, we are using combinations. Combinations help us figure out how many ways we can select items where order does not matter. For example, selecting 3 out of 5 different-colored balls irrespective of the sequence.

• A permutation focuses on arrangement with order, while a combination focuses on selection without regard to order.
• To tackle the card problem, we apply the concept of combinations.

To select 26 out of 52 unique cards where no duplicates are allowed, we use the formula for combinations: \[{}^{52} C_{26} = \frac{52!}{26! (52-26)!}\]The objective here is straightforward selection, not arrangement.

Probability

Probability measures the likelihood that a particular event will occur. It is integral in assessing the possible outcomes of card games and serves as the foundation for understanding random events.

In this exercise, probability aids in understanding the chance of dealing 26 cards without repetition from two decks.

• We start by considering that there are multiple combinations possible when selecting from a deck.
• The goal is to ensure no two cards of the same type are dealt, using combinations to maintain this distinct probability level.

Given two decks, despite having 104 cards, we form our probability model by focusing on the unique 52 types to avoid duplication in 26 picked cards. Ensuring each card picked is different boosts the probability of a flawless selection conforming to the no-duplicate rule.
Understanding probability not only helps deal cards fairly but also improves strategic decision-making in card games.