Patient Age ---> Avg Visits / Year
<1 year ---> 7.5
1-4 years ---> 3.0
5-14 years ---> 1.8
15-24 years ---> 1.7
25-44 years ---> 2.6
45-64 years ---> 4.0
65-74 years ---> 6.7
75 years ---> 8.2
The table above breaks out, by age range, the average number of visits people make to physicians each year. If you were an insurance company that wanted a member mix that would average 50 physician visits per year, you could sign up either of the following two member mixes: 10 members in the "1-4 years" age range and 5 members in the "45-64 years" age range resulting in (10 x 3.0) + (5 x 4.0) = 50
2 members in the "<1 year" age range, 10 members in the "5-14 years" age range and 10 members in the "15-24 years" age range resulting in (2 x 7.5) + (10 x 1.8) + (10 x 1.7) = 50
Note that there are many more possibilities.
Question 1: What is the total number of different member mixes that result in 50 physician visits per year?
Take the co-efficient of $x^{50}$ in the expansion of
$$\frac{1}{(1-x^{1.7})(1-x^{1.8})(1-x^{2.6})(1-x^{3.0})(1-x^{4.0})(1-x^{6.7})(1-x^{7.5})(1-x^{8.2})}$$
or if you prefer working with integers then remove the decimal points and take the co-efficient of $x^{500}$.