How Many Two-digit Numbers Are There, Where The Sum Of The Digit Is Ten?

The sum of two digits can be 1 in 2 ways - 10 or 01. Out of these, 01 is not allowed for the first two digits. Thus, the sum can be 1 for both the first two and the last two digits, in 1 × 2 ways (i.e. 1001 and 1010). Similarly, the sum of both pairs of digits can be 2 in 2 × 3 ways (1102, 1120, 1111, 2002, 2020, 2011) and so on, till we have the sum as 9 for which there are 9 × 10 ways. Now, the sum of 2 digits can be 10 in 9 ways (19, 28, ..., 91). Thus, we have 9 ways on both sides, and hence 9