How to evaluate this multiple integral? [duplicate]
How do I show that $\int_{0<t_1<t_2<…<t_n<t} dt_1…dt_n$ = $\frac{t^n}{n!}$.
For the case of n=2, this was what I got, which is think is wrong but isn’t sure why: $$\int_{0<t_1<t_2<t}dt_1dt_2=\int(\int \mathbb{1}_{0<t_1<t_2<t} \;dt_1)dt_2=\int \mathbb{1}_{0<t_2<t}\;dt_2=\frac{t}{2}$$
Hint: For $n=2$, we can do the following: Let $$ I=\int_{0<t_1<t_2<t} d t_1 dt_2 $$
Then $$ 2I = \int_{0<t_1<t_2<t} d t_1 dt_2+\int_{0<t_2<t_1<t} d t_1 dt_2 = \int_{0<t_1,t_2<t} d t_1 dt_2=t^2 $$ $$ I = \frac{t^2}{2} $$
How could you treat the case of general $n$ similarly?