count records without count(*) & record of row num 3

You don't mention what database this is.

1) There are various tricks you can do with 'Row Number'. This example is for SQL 2008

WITH OrderedOrders AS
(
    SELECT SalesOrderID, OrderDate,
    ROW_NUMBER() OVER (ORDER BY OrderDate) AS 'RowNumber'
    FROM Sales.SalesOrderHeader
)
SELECT *
FROM OrderedOrders
WHERE RowNumber BETWEEN 50 AND 60;

http://msdn.microsoft.com/en-us/library/ms186734.aspx[^]

So, you could create some SQL for your table and show WHERE RowNumber = 3


2) Why you wouldn't want to use the function that is specifically designed for this purpose is beyond me - must be a homework task!

As the other answers have suggested, you'll need to iterate the rows and count them yourself.

If you need to do this purely in SQL (and again, I'm assuming SQL Server) then you could open a cursor and loop through the rows, incrementing a variable and returning the result.