Calculating possible combinations with restrictions

Question

I have five additives that can be mixed into a chemical. Each one can be mixed in at a discrete percentage from 0.02% to 0.22% of the chemical (i.e. 0.02%, 0.03%, 0.04%,...0.22%). Each additive must be present, so mixed in at least 0.02%. The restriction is that the sum of all percentages cannot exceed 0.3%, but the sum can be less than that, so long as each additive is present. Without the restriction the answer is easy. But I can't figure out how to eliminate the number of combinations that would exceed the maximum allowed. If one additive is mixed at 0.22%, then the other four must be 0.02% each and only that.

Thanks!

Answer 1

We must count the number of integer points in a 5-D space ($i=1\cdots 5$) with

$n_i=2\cdots 22$ and $\sum n_i \le 30$

A trivial simplification: let $m_i = n_i-2$, so now we have

$m_i=0\cdots 20$ and $\sum m_i \le 20$

This corresponds to an equilateral 5-D tetraedron (standard simplex). And the number of points if given by the 5-simplex number (generalization of triangular-tetraedral numbers to five dimensions) :

${21 + 5 -1 \choose 5} = 53130 $

(Notice that the problem was eased because the restrictions eactly coincide with the simplex. If the maximum number were 31 or 29 instead of 30, it would have been a little more difficult)

Answer 2

I made a 5 dimension array of the sums and then counted which ones were below 31. In R:

a=b=c=d=e=2:22

all combinations of a + b

ab=outer(a,b,FUN="+")

all combinations of a + b + c

abc=outer(ab,c,FUN="+")

...

abcd=outer(abc,d,FUN="+")

abcde=outer(abcd,e,FUN="+")

So the size of abcde is 21x21x21x21x21 about 4million entries

length(
which(abcde<31)
)

Gives:

53130

If you want the combinations listed use:

mixes=(which(abcde<31,arr.ind=T)+1)/100