Table's rows count without using COUNT() function

You can get table's total number of rows as:

SELECT SUM(ROWS) AS Total_Rows
FROM SYS.SYSINDEXES
WHERE ID=OBJECT_ID('Table1')
AND INDID IN (0,1)

OBJECT_ID('Table1'):
This function will return the object id for table "Table1" or any other specified table.

INDID:
It can be 0 for heap and 1 for clustered index, and it will have only one value at a time 0 or 1, greater than 1 values are for non-clustered indexes.