how many of the integers from 1 to 200 contains the digit 1 at least twice?
Have you written them down and counted them?
Yes i did. The figure that contains the digit 1 twice is only the number 11. I hope my answer is right.
OK I see, thanks you so much.
You're very welcome.
Did you also include 121, 131, etc.?
Yes I did,thanks again Ms.
To find the number of integers from 1 to 200 that contain the digit 1 at least twice, you can use a simple counting method.
First, let's break down the problem into smaller cases:
Case 1: Numbers that have two 1's.
In this case, the 1's can be in any two of the three available digits. The remaining digit can be any of the nine digits (0 to 9) excluding 1. So, there are 3 options for the placement of the two 1's and 9 options for the remaining digit. Thus, there are 3 * 9 = 27 numbers that have exactly two 1's.
Case 2: Numbers that have three 1's.
Here, the three 1's can be placed in any combination of the three available digits. That gives us 3 options. The remaining digit can again be any of the nine digits excluding 1. So, there are 3 * 9 = 27 numbers that have exactly three 1's.
For the uniqueness of numbers, we don't need to consider any cases where the four digits have more than three 1's, as the largest digit we consider is 2.
Therefore, there are 27 numbers with two 1's and 27 numbers with three 1's, giving us a total of 27 + 27 = 54 numbers from 1 to 200 that contain the digit 1 at least twice.
What about?
101
111
112
113
114
etc.