How many three digit numbers can be formed such that the sum of the digits is 6?

Answer 1

21 ways assuming no leading zeros, that is the smallest possible number with three digits is 100

Otherwise if leading zeros are allowed, it is 28 ways.

For three digit numbers the first digit can be 1-6. Leaving the remaining two digits to be 6-value_of_first_digit.

For first digit 6, the remain two digits sum to 0 which means they can only be 00, ie the number 600 - 1 number

For first digit 5, the remain two digits sum to 1 which means they can be 01 or 10, ie the numbers 501, 510 - 2 numbers

For first digit 4, the remain two digits sum to 2 which means they can be 02. 11, 20, ie the numbers 402, 411, 420 - 3 numbers

etc

It can be seen that the number of ways of making a two digit number (with leading zeros) sum to a number less than 10 is the required sum plus 1. So for the remaining first digits, the number of ways are: 3 -> 4 ways, 2 -> 5 ways, 1 -> 6 ways.

Thus the total number of ways is 1 + 2 + .. + 6 = 21

If leading zeros are permitted, so that, for example, 060 (60) and 006 (6) are considered as three digits numbers, then there are a further 7 ways with a first digit of 0, making a total 28 ways.