Problem 36 - Algebra 2

Question

Hachi-hachi is a Japanese game that uses a deck of Hanafuda cards which is made up of 12 suits, with each suit having four cards. How many 7-card hands can be formed so that 3 are from one suit and 4 are from another?

Step-by-Step Solution

Answer

This hand can be formed in 792 ways.

Step 1: Understand the Problem

We need to determine the number of ways to form a 7-card hand from a Hanafuda deck such that 3 cards come from one suit and 4 cards come from another suit. Each suit in the Hanafuda deck contains 4 cards.

Step 2: Select 2 Different Suits

First, choose 2 different suits for the hand. From 12 total suits, the number of ways to choose 2 is calculated by the combination formula: \( \binom{12}{2} \).

Step 3: Choose Cards from the First Suit

For the first chosen suit, you need 3 cards out of the 4 available. The number of ways to choose these cards is \( \binom{4}{3} \).

Step 4: Choose Cards from the Second Suit

For the second chosen suit, you need 4 cards out of the 4 available. The number of ways to choose these cards is \( \binom{4}{4} \).

Step 5: Calculate Total Number of Hands

Multiply the number of ways to choose the suits by the number of ways to choose cards from each suit: \( \binom{12}{2} \times \binom{4}{3} \times \binom{4}{4} \).

Key Concepts

Hanafuda cards

Hanafuda cards are a traditional Japanese card game used for a variety of games such as Koi-Koi and Hachi-hachi. The word 'Hanafuda' translates to 'flower cards.' They are unique as compared to Western playing cards.

• A Hanafuda deck is composed of 12 suits.
• Each suit represents a month of the year, often symbolized by flowers or plants specific to that month correspondingly.
• There are four cards in each suit, making a total of 48 cards in a full deck.

This distinct structure requires players to learn different strategies than those used in games with Western decks. The artistic designs add an element of beauty, making them collectible as well.

combination formula

The combination formula is a fundamental concept in combinatorics, which helps to calculate the number of ways to choose a subset from a larger set. This is critical in determining possible outcomes without regards to the order of selection. The formula is expressed as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where:

• \( n \) is the total number of items to choose from,
• \( k \) is the number of items to choose,
• \( ! \) denotes factorial, which is the product of an integer and all the positive integers below it.

Understanding this formula allows for the calculation of various combinations, such as the selection of cards from different suits in a Hanafuda deck. Using this formula, students can determine the number of ways to form a specific card hand, like in the given exercise.

card suits

In the context of playing cards, the term 'suits' refers to the categories into which the cards are divided. For a Hanafuda deck:

• There are 12 suits, each tied to a month and usually represented by specific flora.
• Unlike the more common four suits seen in Western decks (hearts, diamonds, clubs, spades), Hanafuda's suits are closely linked to nature.
• Each suit's four cards vary in their artwork, often showcasing different aspects of the particular theme or plant.

Understanding the suits in a Hanafuda deck is crucial for playing games like Hachi-hachi, as the strategy relies on which suits and cards are played. Concepts of suits and combinations come together to allow players to form hands, accounting for the number and types of suits involved, as seen in tasks like forming a 7-card hand from the deck.