Finding ways to move from one location to the other

Question

To move from $A$ to the places marked by black dots, if in each step it is only allowed to move right or up, how many ways are there to do this?

Answer 1

Hint: Denote the black dots from top left to bottom right by $i \in \{0, \dots, 6 \}$, e.g. the $0$-th dot is the highest one. Now note that if you want to get to the $i$-th dot, you will always need to do exactly 6 steps, each of which is either up or right.

Suppose you always only walk 6 times up. This will do for the first one, but not for the others: To reach the $i$-th dot, you will need to change $i$ of those "up"-steps to "right"-steps.

Hence the answer should be $\binom{6}{i}$ possibilities to reach $i$-th dot.