Question

The number of positive integer solutions (x,y) such that 3x + 2y = 101​

Answer 1

Answer:

the world of combinatorics, you can use something commonly known as the "stars and bars" method. Generally put, the equation

x1+x2+…+xk=s

where s,xi are positive integers has (s−1k−1) many solutions. The quantity (s−1k−1) is called a binomial coefficient. A general binomial coefficient (ab) where a,b are nonnegative integers with a≥b is defined as

(ab)=a!b!(a−b)!

which means

(s−1k−1)=(s−1)!(k−1)!(s−k)!

In your problem we have s=37 and k=2. So, we want to calculate

(37−12−1)=36!1! 35!=36

However, this tells us that the equation x+y=37 has 36 solutions. Since your equation is 3x+2y=37, we'll want every third x and every second y. This is tantamount to dividing 36 by 3 and 2. We get 363⋅2=6, which matches your calculation. (Had the ratio not been an integer, you'd want to round down to the nearest integer). Depending on how large k and s are, you may find the stars and bars method to be preferable to calculating every solution by hand. It really boils down to a pretty easy factorial calculation. I'd say your problem could go either way.