TchrWill said:
Not when each trip must be 10 blocks long.
Oh come on, 5+5=10.
This is such an old standard question in a city-block metric.
How many paths are there in an m by n grid using the city-block metric, from and to opposite 'corners', making steady progress.
Yes, old and standard, but
The question was "how many different paths can you take from your house TO the store?"
......................................(using a 5x5 square)
.......................Be certain to make your trip (TO THE STORE)
................................. exactly 10 BLOCKS in length
..................................(so only move up or right).
Had the question been "how many different paths can you take from your house, TO the store, and back?", the standard answer would apply.
The general expression that can be derived for the total number of paths from one corner of a rectangular figure, to the diagonally opposite corner, always moving in one of only two possible directions toward the target point, is given by
P = (m + n)!/m!n!
where P is the total number of paths, m is the number of squares on the long side of the rectangle, and n is the number of squares on the shorter side. Of course, ! means factorial. Another way of expressing it it terms of the number of intersections that the paths must connect is
P = [(p - 1) + (q - 1)]!/(p - 1)!(q - 1)!
where p and q are the number of intersections involved on the long and short sides respectively. Thus, for the squares we are addressing:
No. of squares No. of intersections (n + n)! n! P
.......1x1.......................2x2.....................2..........1..........2
.......2x2.......................3x3 ...................24.........2...........6
.......3x3.......................4x4...................720........6..........20
.......4x4.......................5x5.................40320......24..........70
For a rectangle with 11 squares by 4 squares, or 12 by 5 intersections,
P = (11 + 4)!/11!(4!) = 1365 paths.
