Question

An ant starts at position (0, 0) and walks north, east, south or west, one unit at a time.
How many dierent paths of length 8 units finish at (0, 0)?

Answer 1

Take a look at the following article, and see what you can come up with:

https://betterexplained.com/articles/navigate-a-grid-using-combinations-and-permutations/

Answer 2

move 1 unit East: 8!/3!3!=1120
move 2 units East: 8!/2!2!2!2!=2520
move 3 units East: 8!/3!3!=1120
move 4 units East: 8!/4!4!=70
move 0 unit East: 8!/4!4!=70
Total paths: 4900

Note: if ant moves 1 unit East, must move back 1 unit West.