{My 100th problem} Sum of digits =5?
Number Theory
Level
3
How many positive integers less than 10,000 are there in which the sum of the digits equals 5?
62
31
56
93
51
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1 solution
Let us try some examples, 5, 1211, 230. As the numbers must smaller than 10000, it at most has 4 digits.
Look at the examples again, 5=0005 (which means 5=0+0+0+5), 1211(5=1+2+1+1), 230=0230(5=0+2+3+0). We can find that every unique number means a unique partition of 5. But digits of numbers have specific position (such as 312 and 321 are two different numbers.) Hence, the answer of this question is same as the way of partition of 5 in four integers considering its position (5=0+0+0+1 and 5=0+1+0+0 are consider different .)
This is actually as same as ways of putting 5 identical balls into four different basket. Which is equal to ( 5 5 + ( 4 − 1 ) ) = 5 6 .
(If you are interested why it is equal to ( 5 5 + ( 4 − 1 ) ) = 5 6 , search 'identical object into distinct bins' in Brilliant)