How do i find t given 3 = t!+(t+1)!/t!+t!

Question

I am a high school student, so I apologize for what might be a simple question. The original question is:

There are an equal number of boys and girls in a class and they are seated in a row so that no two boys sit next to each other and no two girls sit next to each other. If one more student is added to the class the number of seating arrangements with the original conditions is tripled. How many students were in the class originally?

from this I wrote 3(t!+t!) = t!+(t+1)!, but the problem is I have no idea how to evaluate and solve for t. Thanks for your time!

Answer 1

Assertion: $$3(t!+t!)=t! + (t+1)!$$

---

Evaluation: $$\begin{align} 3\cdot 2t! = 6t! &= t! + (t+1)! \\ 5t! &= (t+1)! \\ 5 &= \frac{(t+1)!}{t!} \\ &= t+1 \\ t &= 5-1 \end{align}$$

---

Answer: $$\boxed{t=4}$$

---

I recently got a new phone with a new iOS so tell me if the formatting of my answer is not as it should be, and I will fix it.

Answer 2

The counting is off. Note that for each of the $t!$ arrangements of boys, there are $t!$ arrangements of girls. Thus there are $2t!t!$ different arrangements, where we multiply by $2$ to account for whether we start with boys or girls.

Thus the equation becomes $$3(2(t!)^2)=t!(t+1)!,$$ where there is no $2$ on the right hand side because whichever gender with $t+1$ members must appear first and last. To solve this, divide by $(t!)^2$ to get $6=t+1$, and therefore $t=5$.