How many four digit numbers have the property that their digits taken from left to right form an arithmetic or a geometric progression?

The numbers forming an AP would be: 1234, 1357, 2345, 2468, 3210, 3456, 3579, 4321, 4567, 5432, 5678, 6543, 6420, 6789, 7654, 7531, 8765, 8642, 9876, 9753, 9630. A total of 21 numbers.

What is the number of 3 digit numbers with distinct digits in which digits are in AP?

3rd place = 3 options. Now, to find all the arrangements, we multiply the number of options in all places. Thus there are 60, 3 digit numbers, with distinct digits, with each of the digits odd.

How many numbers can you get from 3 digits?

There are, you see, 3 x 2 x 1 = 6 possible ways of arranging the three digits. Therefore in that set of 720 possibilities, each unique combination of three digits is represented 6 times.

How many 3 digit numbers have distinct and nonzero digits?

Explanation: D In a 3-digit number containing no zeros, there are nine possibilities for the first digit (1-9); nine possibilities for the second digit (1-9); and nine possibilities for the third digit (1-9). That makes a total of 9 × 9 × 9 possible 3-digit numbers, or 729.

What is the smallest 3-digit number with distinct digits?

102
Thus, 102 is the smallest 3-digit number with unique digits.

How many three-digit numbers that satisfy the middle digit property?

If the middle digit is , the only possible number is . So there is number in this case. So the total number of three-digit numbers that satisfy the property is Alternatively, we could note that the middle digit is uniquely defined by the first and third digits since it is half of their sum.

What is the sum of the first and third digits?

If both the first and last digits are even, then we have 2, 4, 6, 8 as our choices for the first digit and 0, 2, 4, 6, 8 for the third digit. There are numbers here. As we noted in Solution 2, we note that the sum of the first and third digits has to be even.

How many arithmetic progressions are there with common difference 0?

Now let’s count the arithmetic progressions. There are the 9 with common difference 0. There are 7 increasing ones with common difference 1, and 8 decreasing ones, since 0 can be the final digit in that case. There are 5 increasing ones with common difference 2, and 6 decreasing ones.

What are the numbers if the first and last digit is odd?

If both the first digit and the last digit are odd, then we have 1, 3, 5, 7, or 9 as choices for each of these digits, and there are numbers in this case. If both the first and last digits are even, then we have 2, 4, 6, 8 as our choices for the first digit and 0, 2, 4, 6, 8 for the third digit.