How many sequences of length 5 can be formed using the digits 0, 1, 2, . . . , 9 with the property that exactly two of the 10 digits appear, e.g., 05550

Hi Guests, asker and answerer.

Here is my take. No promises that I am right though.

There are 10C2 = 45 conbinations of 2 numbers

You can have them in the ratio 1:4 or 2:3

It you have 1 of one of the numbers and 4 of the other then there is 5 ways to do that.

But either can be the single so that is 2 times

so I have 2*5*45 = 450 ways

If I have 2 or 1 and 3 of the other then i have 2*5C2ways = 20

20*45=900 ways

900+450 = 1350 combinations that meet that requirement.