# Solve an equation which contains binomial coefficient

I have the following equation: $${{x}\choose{3}}=10$$ I want to solve it for $x$ so I wrote: $${{x}\choose{3}}=\frac{x!}{3!(x-3)!}=10$$ which follows: $$\frac{x!}{(x-3)!}=60$$ I wonder how to proceed from here and solve for $x$? As $(x-3)!\neq x!-3!$.

• Think how to further simplify the left-hand side with the division. Plug in the definition of the factorial. Commented May 11, 2018 at 10:44
• Would you accept $-1 \pm\sqrt{-11}$? Commented May 11, 2018 at 15:55

Hint: Simplifying $$\binom{x}{3}=\frac{x(x-1)(x-2)}{1\cdot 2\cdot3}$$ we get $$\frac{x^3-3x^2+2x}{6}=10$$

$$x!=1\cdot2\cdot3\cdot(...)\cdot(x-3)\cdot(x-2)\cdot(x-1)\cdot x$$ $$3!(x-3)!=6\cdot1\cdot2\cdot3\cdot(...)\cdot(x-3)$$ Hence $$\frac{x!}{3!(x-3)!}=\frac{x(x-2)(x-1)}{6}=10$$ and solve from there.

$$\frac{x!}{(x-3)!} = x(x-1)(x-2)$$
as $x(x-1)(x-2)=60$ which is a cubic equation it can easily be solved
giving $x=5$

Observe that $$\binom{x}{3} = \frac{(x)(x-1)(x-2)}{6}$$

$$\frac{(x)(x-1)(x-2)}{6} = 10 \implies (x)(x-1)(x-2)-60 = 0$$ $$x^3 -3x^2+2x-60 = 0$$

You can factorise the above cubic equation to finish off