Problem Solving

hey guys,

I dont really get the answer to question no. 38 (especially the donut thing):


38. How many positive integers less than 10,000 are there in which sum of digits equals 5?
(a) 31
(b) 51
(c) 56
(d) 62
(e) 93



Solution:


Here, we have 4 digits (positive integer less than 10000 means that 9999 is the biggest and we can pretend that "5" is "0005") that must sum to 5.
Since we have 4 digits, we'll have 3 partitions. We're summing to 5, so we have 5 "donuts".
O O O O O

Since we can use 0, we can have multiple partitions in the same spot. For example, we could have:
|||OOOOO (which translates to 0005)
we could have:
||O|OOOO (which translates to 0014, or 14).

So, we view this as a permutation question: we have 8 total objects, 3 of which are identical to each other (the partitions) and 5 of which are identical to each other (the donuts). Using the permutation formula for which some objects are identical:
Total permutations = n!/r!s! = 8!/3!5! = 8*7*6/3*2*1 = 8*7 = 56... Choose (C).