Basic Discrete Mathematics Question

Question

I was preparing my self before an exam and I found this question:

For each of the following equations, find a positive integer $n$ that satisfies the equation. The notation $p(n,r)$ stands for $n(n-1)\ldots(n-r+1).$

$p(n,2) = 30$

$p(n,3)=24p(n,2)$

$10p(n,2)=p(3n-1,2)+40$

How can I solve this?

Any help will be appreciated!

Answer 1

Presumably this is not a single problem, but three separate problems. The notation $p(n,r)$ most likely stands for $n(n-1)(n-2)\ldots(n-r+1),$ the number of ways to fill $r$ slots with elements taken from a set of size $n.$ Each exercise then boils down to some algebra.

Answer 2

Once you expand the $p$ notation, these problems ask you to solve the following equations for $n$, where $n$ is a positive integer:

1. $n(n-1)=30$
2. $n(n-1)(n-2)=24n(n-1)$
3. $10n(n-1)=(3n-1)(3n-2)+40$

The first can easily be solved by eye, and the second almost as easily; in neither case should you have to set and solve an equation. You will probably have to do so for the third, but the equation is only a quadratic and therefore easy to solve.