Find the number of 8 digit numbers sum of whose digits is 4.
Suppose the number is .The possible values of are 1, 2, 3, 4. We consider these four cases.
If then all other digits are 0 (since sum of digits is 4). Hence there is only 1 such number.
If then exactly one of the other 7 digits is 1. Hence there are 7 such numbers (depending on where the digit '1' is).
If $latex a_1 = 2 $ then sum of the other seven digits is 2.
Hence we compute the number of non negative integer solutions of $latex a_2 + ... + a_8 = 2 $ .
This equals = 28
If then sum of the other seven digits is 3.
Hence we compute the number of non negative integer solutions of = 3
This equals = 84
Hence the answer is 120.
For more problems: Pre-Regional Mathematics Olympiad Problems