Problem: How many integers greater than 53,000 have the following two properties: the digits of each integer are distinct; the digits 8 and 9 do not occur in any integer.
Answer: 90,360.
Case #1: the integer is 8 digits
The first digit cannot be 0, 8, 9 so there are 7 choices. The second can be a 0 or any digit that was not chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 7 x 7! = 35,280
Case #2: the integer is 7 digits
The first digit cannot be 0, 8, 9 so there are 7 choices. The second can be a 0 or any digit that was not chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 7 x 7 x 6 x 5 x 4 x 3 x 2 = 35,280
Case #3: the integer is 6 digits
The first digit cannot be 0, 8, 9 so there are 7 choices. The second can be a 0 or any digit that was not chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 7 x 7 x 6 x 5 x 4 x 3 = 17,640
Case #4: the integer is 5 digits
This needs to be broken into 2 sub cases: first digit is 5, and the first digit is greater than 5.
Case 4-a: first digit is 5
The first digit is 5 so is 1 choice. The second digit cannot be a 5, 1, 2, 0, 8, 9 so there are 4 choices. The third digit cannot be a 5, 8, 9, or any digit that was chosen second so there are 6 choices. Each successive digit has one less choice than the previous digit. 1 x 4 x 6 x 5 x 4 = 480
Case 4-b: first digit is 6 or 7, so there are 2 choices. The second cannot be 8, 9, or any digit that was chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 2 x 7 x 6 x 5 x 4 = 1,680
35280 + 35280 + 17640 + 480 + 1680 = 90,360 total integers.
Answer: 90,360.
Case #1: the integer is 8 digits
The first digit cannot be 0, 8, 9 so there are 7 choices. The second can be a 0 or any digit that was not chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 7 x 7! = 35,280
Case #2: the integer is 7 digits
The first digit cannot be 0, 8, 9 so there are 7 choices. The second can be a 0 or any digit that was not chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 7 x 7 x 6 x 5 x 4 x 3 x 2 = 35,280
Case #3: the integer is 6 digits
The first digit cannot be 0, 8, 9 so there are 7 choices. The second can be a 0 or any digit that was not chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 7 x 7 x 6 x 5 x 4 x 3 = 17,640
Case #4: the integer is 5 digits
This needs to be broken into 2 sub cases: first digit is 5, and the first digit is greater than 5.
Case 4-a: first digit is 5
The first digit is 5 so is 1 choice. The second digit cannot be a 5, 1, 2, 0, 8, 9 so there are 4 choices. The third digit cannot be a 5, 8, 9, or any digit that was chosen second so there are 6 choices. Each successive digit has one less choice than the previous digit. 1 x 4 x 6 x 5 x 4 = 480
Case 4-b: first digit is 6 or 7, so there are 2 choices. The second cannot be 8, 9, or any digit that was chosen first, making a total of 7 choices. Each successive digit has one less choice than the previous digit. 2 x 7 x 6 x 5 x 4 = 1,680
35280 + 35280 + 17640 + 480 + 1680 = 90,360 total integers.
