Single Sentenced Problem!
In how many different ways can the number 2015 be written as a sum of three positive integers, not necessarily distinct?
Bonus: Generalize the above problem for integers .
Note: Sums like and , etc. are considered to be the same .
The answer is 338352.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
One may start with calculating the integrated quantity of ordered solutions of the three positive numbers(say a, b, c), which is 2015-3+3-1 choose 3-1 = 2027091(stars and bars theorem). Now we can exclude the possibilities where two variables are identical in order to obtain a clean set of variable-wide, mutually distinct solutions. To do this, study the equation 2k+l = 2015, that has one degree of freedom. There are floor(2015/2) = 1007 solutions, and by considering the choices of k, there are 1007*(3 choose 2) = 3021 solutions. Therefore, there exist 2027091-3021=2024070 solutions for mutually distinct a, b, c. Due to the remark in Note , we have to divide 2024070 by 3! and get 337345. Lastly, adding 1007, which is the number of solutions of unordered a, b, c where two variables are identical, we acquire the final answer of 338352.