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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
There are 4500 four-digit odd numbers in which the leftmost digit is not zero.

Step by step solution

01

Determine the Number of Choices for Each Digit

A four-digit number has four places that need to be filled. From the left, these are the thousands, hundreds, tens, and ones places. Since the digit on the left cannot be zero, there are 9 choices (from 1 to 9) for the thousands place. For the hundreds and tens places, all 10 digits (from 0 to 9) can be used so there are 10 choices for each. The ones place determines whether a number is odd or even. For a number to be odd, it must end in 1, 3, 5, 7, or 9, giving 5 choices for the ones place.
02

Calculate the Total Number of Four-Digit Odd Numbers

The total number of four-digit odd numbers can be found by calculating the product of the number of choices for each digit. So, multiply the number of choices for the thousands place (9), hundreds place (10), tens place (10), and ones place (5) together: \(9 \times 10 \times 10 \times 5\).
03

Compute the Final Result

When you perform the multiplication \(9 \times 10 \times 10 \times 5\), you get a result of 4500. This means there are 4500 different four-digit odd numbers where the leftmost digit is not zero.

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