Since you are looking for 2-digit combinations rather than 2-digit numbers I would include combinations such as 02, 09 and even 00 as 2-digit combinations. You can list them all if you wish using the "natural order" which you might think of as the numeric order of dictionary order
00, 01, 02, ..., 99
Gary wrote back,
Maybe I asked the wrong question based on the answer I received I am looking for all the 2 digit numbers using number 0 thru 9 thanks Gary
In this case I would not include 02, 09, 00 or any 2-digit combination that has 0 in the leftmost place. The 2-digit numbers, again in the natural order are
10, 11, 12, ..., 99
Gary wrote back,
maybe again i asked wrong question ie 223 36 67 93 54 etc Gary
So let me try again. It looks like you want all positive integers that contain only two distinct digits. For example one 5 digit positive integer containing the digits 3 and 7 is 37737. I am going to restrict my response to 5 digit positive integers and you can generalize to other choices for the number of digits.
Suppose I want all 5 digit positive integers made from the digits 3 and 7. They could have
one 3 and four 7s
two 3s and three 7s
three 3s and two 7s or
four 3's and one 7.
There are ways to choose 1 position (for the 3) from 5 positions and hence there are five 5 digit positive integers with one 3 and four 7s, they are
There are ways to choose 2 positions (for the 3s) from 5 positions and hence there are ten 5 digit positive integers with two 3s and three 7s, they are
That's 15 so far and you can get another 15 by interchanging the 3s and 7s, for a total of 30.
so when n = 5
= 25 - 2 = 32 - 2 = 30
What about digits other than 3 and 7? If you don't count zero then you can choose a first digit to include in 9 ways and then a second digit in 8 ways. Thus there are 9 8 ways to choose two digits. If zero is one of the digits then there are 9 ways to choose the second digit but half of the "five digit" numbers you form have a zero in the first place which makes them a four digit number. Thus the total number of five digit numbers that can be formed using two distinct digits is
9 8 (25 - 2 ) + 9 (25 - 2 )/2