Among the integers from 1 to 10, 000, 000, 000 which are there more of: Those in
which the digit 1 occurs or those in which it does not occur?
number without a 1
= 9^7
= 4,782,969
so number where no 1 occurs = 4,782,969
number where a 1 does occur = 10,000,000-4,782,969
= 5,217,031
how do you prove this?
To determine whether there are more integers between 1 and 10,000,000,000 that contain the digit 1 (inclusive) or do not contain the digit 1, we can use mathematical reasoning.
First, let's consider the integers that do not contain the digit 1. We can break down this group into two categories: integers with fewer than 10 digits and integers with 10 digits.
Category 1: Integers with fewer than 10 digits
- There are 9 possible digits for each position (0-9), and since we want to exclude the digit 1, there are 9 options for each digit position.
- For each digit position, there are 9 options, so the total number of integers with fewer than 10 digits that do not contain the digit 1 is 9^9.
Category 2: Integers with 10 digits
- Since we want to exclude the digit 1, there are 9 options for the first digit (0-9, excluding 1).
- For the remaining 9 digits, each position has 10 options (0-9). Therefore, the total number of integers with 10 digits that do not contain the digit 1 is 9 * 10^9.
Now, let's consider the integers that contain the digit 1. We can break down this group into two categories as well: integers with fewer than 10 digits and integers with 10 digits.
Category 3: Integers with fewer than 10 digits
- For each digit position, there are 10 options (0-9). However, we have to subtract 1 option which is the absence of the digit 1.
- Therefore, the total number of integers with fewer than 10 digits that contain the digit 1 is 10^9 - 1.
Category 4: Integers with 10 digits
- For each digit position, there are 10 options (0-9). Therefore, the total number of integers with 10 digits that contain the digit 1 is 10^10.
To find out which group (those with digit 1 or those without digit 1) has more integers, we can simply compare the sum of the integers in each group.
Sum of integers without digit 1 = 9^9 + (9 * 10^9)
Sum of integers with digit 1 = (10^9 - 1) + 10^10
By comparing these two sums, you can determine which group has more integers.
Note: Calculating the actual values of these sums may require using a calculator or programming to handle large numbers.