Selection of Two Cards of the Same Suit from a 36-Card Deck

The problem of selecting two cards of the same suit from a 36-card deck is a classic example in probability theory. In this article, we will explore the correct methods to solve this problem and provide the number of ways this can be done.

Understanding the Problem

The original problem statement is somewhat ambiguous. To ensure clarity, we assume a standard configuration where there are 4 suits (clubs, spades, hearts, diamonds) and each suit contains 9 cards. This is a common setup for games like Piquet or Trappola, where a stripped deck of 36 cards is used.

Calculation of Selection Ways

We need to determine how many ways two cards of the same suit can be selected from a deck of 36 cards, where each suit has 9 cards. Let's break this down step by step.

Method 1: Combinatorial Approach

Given that there are 4 suits, each with 9 cards, we can use the combination formula to calculate the number of ways to select 2 cards from 9 for one suit and then do the same for all suits:

For a given suit, the number of ways to choose 2 cards out of 9 is given by the combination formula ( binom{n}{k} ), where ( n ) is the total number of cards in the suit (9 in this case) and ( k ) is the number of cards to be chosen (2 in this case).

The formula for combinations is:

[ binom{n}{k} frac{n!}{k!(n-k)!} ]

Substituting ( n 9 ) and ( k 2 ):

[ binom{9}{2} frac{9!}{2!(9-2)!} frac{9 times 8}{2 times 1} 36 ]

Since there are 4 suits, the total number of ways to select two cards of the same suit is:

[ 4 times 36 144 ]

Method 2: Direct Counting Approach

Another way to look at this problem is to directly count the number of favorable outcomes. For each card selected from the deck, there are 8 remaining cards in the same suit:

1. The first card can be any of the 36 cards.

2. The second card must be one of the 8 remaining cards in the same suit.

However, since we are counting pairs, each pair is counted twice (once for each order). Therefore, we need to divide the product by 2:

[ 36 times 8 288 ]

[ frac{288}{2} 144 ]

This confirms our earlier calculation.

Conclusion

In summary, if a deck contains 36 cards with 9 cards in each of the 4 suits, the number of ways to select 2 cards of the same suit is 144.

It's important to note that the problem could be more complex if different assumptions are made, such as the deck having fewer or more suits, or different numbers of cards per suit. Always ensure the problem statement is clear and specific to avoid confusion and incorrect solutions.

Keywords

deck selection, probability, same suit selection