Logan is travelling to school from his home which is 7 blocks west and 3 blocks south of the
school. If he always walks in the direction of the school, how many different paths could he take?
To determine the number of different paths Logan could take, we can use the concept of permutations.
Since Logan needs to travel 7 blocks west and 3 blocks south, we can think of his path as a sequence of movements. Let's represent moving west as "W" and moving south as "S". Therefore, Logan's path can be represented as a sequence of 7 "W"s followed by 3 "S"s.
In this case, we have a total of 10 movements (7 "W"s + 3 "S"s), and we need to arrange them in a specific order. The number of different paths can be calculated using the formula for permutations:
nPr = n! / (n - r)!
Where n is the total number of movements (10) and r is the number of movements in one direction (7 "W"s).
Applying the formula:
nPr = 10! / (10 - 7)!
= 10! / 3!
Now, we can calculate the factorial values:
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
3! = 3 x 2 x 1
Plugging these values back into the formula:
nPr = (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (3 x 2 x 1)
Simplifying the expression further:
nPr = (10 x 9 x 8 x 7 x 6 x 5 x 4) = 604,800
Therefore, Logan could take a total of 604,800 different paths to get to school.