How many ways are there to go from the point $(0,0)$ to $(m,n)$ using,right up and down moves that we don't pass a point more than once? How many ways are there to go from the point $(0,0)$ to $(m,n)$ using,right up and down moves that we don't pass a point more than once?
I tried using to calculate every case(depending on the down moves) then maybe the identity $\sum\limits_{r=0}^m n^r \binom{m}{r}=(n+1)^m$ will work.(because the answer in the book is $(n+1)^m$).But it didn't work.Any hints?

Under the assumption that movement is only allowed in the grid formed by corner points $(0,0), (m, 0), (0, n), (m,n)$:
For each column, there are $n+1$ ways to pick where the horizontal movement occurs. After placing the horizontal movements for each column, there's only one way to connect them all (and these connections are done entirely in terms of up/down movements that do not revisit any position more than once).
Over all $m$ columns, that's $(n+1)^m$ paths in total, which matches the answer in your book.
