How many 4-digit positive integers are there, where each

Every digit has to be positive.. So "0" would not be valid... In 1212, the second digit can only be 2,3,4,5,6,7,8 or 9...

First digit takes numbers from 1 to 9 --- 9 nos
second digit onwards 0 to 10 --- 10 nos......but to avoid repetition of the previous one ----- 9 nos

hence total nos = 9*9*9*9 = 6561

Just to elaborate a little since there was a bit of confusion up there:

We need 4-digit numbers, where each digit is positive. There are 9 positive digits.

Starting from right to left, we have:

9 possibilities *
(9 - the digit just used = 8 possibilities) *
(Again, 9 - the digit just used) = 8 possibilities) *
(Again, 9 - the digit just used) = 8 possibilities)
= 9 * 8 * 8 * 8 = 4608.

If anyone is thrown off by the fact that if the first and third numbers are different, the second digit only has 7 possibilities, this should clarify:

Suppose we didn't choose numbers from left to right, and consider the problem in the following way:

-(1)- -(2)- -(3)- -(4)-

(1) has 9 possibilities; nothing has been chosen
(3) has 9 possibilities; nothing has been chosen
(4) has 8 possibilities; it cannot be the same as (3).
(2) now has 7 possibilities right? since (1) and (3) could be different? Nope! .. Well, sort of.

Here, we have to split the problem into two scenarios:
Case A: 1 and 3 have the same digit --> there are 8 options for the second digit.
Case B: 1 and 3 have different digits --> there are 7 options for the second digit.

So, adding up the two cases, we get: (Case A: 9 * 8 * 1 * 8) + (Case B: 9 * 7 * 8 * 8) = 9 * 8 * 8 * 8 = again, 4608.

Same logic applies to any way that you want to consider the problem.

Apologies if that was obvious to everyone else; I was thrown off for a second before realizing how it worked, so thought I'd post just in case anyone else had the same confusion.

Ah that makes sense. thank you.

What is the source of this question?

Posted from my mobile device

why we are not including 0 in second and third place

Because you have been asked to use only positive digits. 0 is neither negative nor positive.

I got some different figure. My approach is as follows: There can be 4 scenarios.
First, 1212----> 9*8*1*1 = 72
Second, 1534-----> 9*8*7*6 = 3024
Third, 1231-------> 9*8*7*1 = 504
Fourth, 2131------> 8*9*7*1 = 504
Total= 72+3024+504+504 = 4104. I know this is wrong. Kindly shed some light on it.

Thank You in advance.

You forgot a case:
Fifth, 2325 (1st and 3rd digits are the same, other two distinct) ---------> 9*8*1*7 = 504

Now the total is 4104 + 504 = 4608

But note that this is a very convoluted approach. Try to understand the simpler approach mentioned above.