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Chapter 11: Problem 30

How many four-digit odd numbers are there? Assume that the digit on the left cannot be 0 .

Short Answer

Expert verified
The total number of four-digit odd numbers is 4500.

Step by step solution

01

Understand the conditions for each digit

Since this is a four-digit number, it should consist of a thousands place, a hundreds place, a tens place, and a ones place. The digit on the left (thousands place) cannot be zero, and the ones place should be odd to make the whole four-digit number odd. An odd number in the ones place can be any of the five odd numbers: 1, 3, 5, 7, 9.
02

Count the possible numbers for each place

For the leftmost digit (thousands place), it can be any digit from 1 to 9, giving a total of 9 choices. For each of the hundreds place and tens place, they can be any digit from 0 to 9, giving 10 choices each. For the ones place, there are 5 choices: 1, 3, 5, 7, 9.
03

Calculate the total number of four-digit odd numbers

Using the Multiplication Rule in counting, which says that if one event can occur in M ways and a second can occur independently of the first in N ways, then the two events together can occur in M x N ways, there are altogether 9 x 10 x 10 x 5 = 4500 possible four-digit odd numbers.

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Key Concepts

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

Understanding Four-Digit Numbers
A four-digit number is made up of thousands, hundreds, tens, and ones places. Four-digit numbers range from 1000 to 9999.
Their internal structure means there is a specific condition: the leftmost digit (thousands place) cannot be zero.
This is because if it were zero, the number would not be a true four-digit number; it would instead be a three-digit figure.
  • Thousands place: The digit here must be from 1 to 9. This makes the smallest number 1000, and the largest 9999.
  • Other digits (hundreds, tens, and ones places): These can range from 0 to 9, allowing a great variety of combinations.
Fully grasping this helps in structuring numbers correctly and determining the criteria for creating any specific set of numbers, such as odd numbers.
Identifying Odd Numbers
Odd numbers are integers not divisible by 2. They end in 1, 3, 5, 7, or 9. For a number to be considered odd, its ones place must be occupied by these digits.
This is a crucial property when generating odd numbers.
  • Example of odd numbers: 1001, 2053, or 9875.
To generate a four-digit odd number, focus on ensuring the ones place is filled by one of these odd digits.
This principle simplifies the task of counting or forming odd numbers when combined with the numbers in other places.
The Multiplication Rule in Counting
The Multiplication Rule is a fundamental principle in combinatorics used to calculate probabilities or the number of possible outcomes.
It states that if one event can occur in \(M\) ways and a second can occur in \(N\) ways, independent of the first, then the two events together can occur in \(M \times N\) ways.

Application to Counting Four-Digit Odd Numbers

In the context of four-digit odd numbers:
  • Thousands place: Can have any digit from 1 to 9, giving 9 possibilities.
  • Hundreds and tens places: Each can have any digit from 0 to 9, providing 10 choices for each place.
  • Ones place: Must have an odd digit, so there are 5 choices (1, 3, 5, 7, 9).
Thus, using the multiplication rule, the total number of combinations is calculated as:\[9 \times 10 \times 10 \times 5 = 4500\]This complete steps illustrate how independent choices lead to multiple combinations using the multiplication rule.

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Most popular questions from this chapter

Involve computing expected values in games of chance. For many years, organized crime ran a numbers game that is now run legally by many state governments. The player selects a three-digit number from 000 to 999 . There are 1000 such numbers. A bet of \(\$ 1\) is placed on a number, say number 115 . If the number is selected, the player wins \(\$ 500\). If any other number is selected, the player wins nothing. Find the expected value for this game and describe what this means

Involve computing expected values in games of chance. A game is played using one die. If the die is rolled and shows 1 , the player wins \(\$ 1\); if 2 , the player wins \(\$ 2\); if 3 , the player wins \(\$ 3\). If the die shows 4,5 , or 6 , the player wins nothing. If there is a charge of \(\$ 1.25\) to play the game, what is the game's expected value? What does this value mean?

Write a probability problem involving the word "and" whose solution results in the probability fractions shown. \(\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}\)

An architect is considering bidding for the design of a new museum. The cost of drawing plans and submitting a model is \(\$ 10,000\). The probability of being awarded the bid is \(0.1\), and anticipated profits are \(\$ 100,000\), resulting in a possible gain of this amount minus the \(\$ 10,000\) cost for plans and a model. What is the expected value in this situation? Describe what this value means.

The probability that a region prone to flooding will flood in any single year is \(\frac{1}{10}\). a. What is the probability of a flood two years in a row? b. What is the probability of flooding in three consecutive years? c. What is the probability of no flooding for ten consecutive years? d. What is the probability of flooding at least once in the next ten years?
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