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Chapter 11: Problem 52

Five cards are marked with the numbers \(1,2,3,4,\) and \(5,\) shuffled, and 2 cards are then drawn. How many different 2 -card hands are possible?

Short Answer

Expert verified
10 different 2-card hands are possible.

Step by step solution

01

Understand the problem

We need to find out how many different combinations of 2 cards can be drawn from a set of 5 cards labeled 1 to 5. The order in which the cards are drawn does not matter.
02

Identify the formula

For combinations where order does not matter, the formula for combinations is used: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] Here, \( n \) is the total number of items, and \( k \) is the number of items to choose.
03

Apply the values

Here, \( n = 5 \) (total cards) and \( k = 2 \) (cards to be drawn). Substitute these values into the combination formula: \[ C(5, 2) = \frac{5!}{2!(5-2)!} \]
04

Calculate the factorials

Factorial of a number \( x! \) means the product of all positive integers up to \( x \). Calculate the factorials: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] \[ 2! = 2 \times 1 = 2 \] \[ 3! = 3 \times 2 \times 1 = 6 \]
05

Substitute and solve

Substitute the factorial values back into the combination formula: \[ C(5, 2) = \frac{120}{2 \times 6} = \frac{120}{12} = 10 \]
06

Conclusion

The number of different 2-card hands that can be drawn from a set of 5 cards is 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics that focuses on counting, arrangement, and combination of objects. It helps us determine the number of different ways to choose or arrange items under given conditions. This is particularly useful in fields such as probability, statistics, and computer science. In our given exercise, we use combinatorics to find the number of possible 2-card hands from a deck of 5 cards.
Factorials
A factorial, denoted by the symbol \(!\), is a mathematical operation that multiplies a number by every positive integer less than itself. For example, the factorial of 5 (written as \(5!\)) is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120.\). Factorials grow very quickly with larger numbers and are extensively used in combinatorics. Here are some essential points:
  • The factorial of 0 is defined as 1: \(0! = 1\).
  • Factorials are only defined for non-negative integers.
In our problem, we use the factorial concept to solve the combination formula.
Combination Formula
The combination formula is used to calculate the number of ways to choose \(k\) items from \(n\) items without considering the order. It is given by \[C(n, k) = \frac{n!}{k!(n-k)!}\]. The logic behind this formula is to divide the total number of arrangements \(n!\) by the arrangements of the selected items \(k!\) and the arrangements of the remaining items \((n - k)!\). Let's break this down with an example:
  • You have 5 cards and need to draw 2 cards. So, \(n = 5\) and \(k = 2\).
  • Using the formula: \[C(5, 2) = \frac{5!}{2! \times 3!} = \frac{120}{2 \times 6} = 10\].
This gives us the final answer: there are 10 different ways to draw 2 cards from a set of 5.

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Most popular questions from this chapter

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