Solve for y $\frac{y!}{(y-2)!2!}+10y=108$

Question

I did $$\frac{y!}{(y-2)!2}+10y=108\\$$

I don't really know how to go from here...

This equation is actually part of a system:

$$\begin{cases}{}^xC_2 = 45 \\ {}^yC_2 + xy = 108\end{cases}$$

Solving the first part of the system I find that x = 10.

My questions:

• How do I solve $\frac{y!}{(y-2)!2!}+10y=108$
• Would it be simpler to solve x and y by solving the system, instead of the isolated equations?
• How do I solve the system?

NOTE: C here is the C from the formula of combinations, as in: $${}^nC_k = \frac{n!}{(n-k)!k!}$$

Answer 1

Assuming you meant to solve $\frac{y!}{(y-2)!2!}+10y=108$,

$$\frac{y!}{(y-2)!}=y(y-1)=y^2-y$$

$$\frac{y!}{(y-2)!2!}+10y=0.5y^2+9.5y=108$$

$$y^2+19y=216$$

Quadratic, with solution given as

$$y=8,-27$$

Usually $y>0$, so then $y=8$.

Answer 2

$$y^2+19y=216$$ $$y(y+19)=(8)(27)$$ so the $$y=8$$