Number of positive integral solutions of the equation x/3 + 14/y = 3, is?
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Answer:
In the world of combinatorics, you can use something commonly known as the "stars and bars" method. Generally put, the equation
x1+x2+…+xk=s
x1+x2+…+xk=s
where s,xis,xi are positive integers has (s−1k−1)(s−1k−1) many solutions. The quantity (s−1k−1)(s−1k−1) is called a binomial coefficient. A general binomial coefficient (ab)(ab) where a,ba,b are nonnegative integers with a≥ba≥b is defined as
(ab)=a!b!(a−b)!
(ab)=a!b!(a−b)!
which means
(s−1k−1)=(s−1)!(k−1)!(s−k)!
(s−1k−1)=(s−1)!(k−1)!(s−k)!
In your problem we have s=37s=37 and k=2k=2. So, we want to calculate
(37−12−1)=36!1! 35!=36
(37−12−1)=36!1! 35!=36
However, this tells us that the equation x+y=37x+y=37 has 3636 solutions. Since your equation is 3x+2y=373x+2y=37, we'll want every third xx and every second yy. This is tantamount to dividing 3636 by 33 and 22. We get 363⋅2=6363⋅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 kk and ss 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.
Step-by-step explanation:
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