Answer:
84 possible paths
Step-by-step explanation:
Given
[tex]N =3[/tex] --- 3 blocks north
[tex]E = 6[/tex] --- 6 blocks east
Required
Number of distinct path
To solve this question, we make use of the following formula
[tex]^{m+n}C_n \ =\ ^{m+n}C_m[/tex]
The above formula implies that;
On a single path, there is a total of m + n steps to get to a particular position, where each path is either in m direction or n direction.
In this case:
[tex]m = 3[/tex]
[tex]n = 6[/tex]
So, we have:
[tex]^{m + n}C_n = ^{3+6}C_6[/tex]
[tex]^{m + n}C_n = ^9C_6[/tex]
Apply combination formula
[tex]^{m + n}C_n = \frac{9!}{(9-6)!6!}[/tex]
[tex]^{m + n}C_n = \frac{9!}{3!*6!}[/tex]
Expand the numerator
[tex]^{m + n}C_n = \frac{9*8*7*6!}{3!*6!}[/tex]
[tex]^{m + n}C_n = \frac{9*8*7}{3!}[/tex]
Expand the denominator
[tex]^{m + n}C_n = \frac{9*8*7}{3*2*1}[/tex]
[tex]^{m + n}C_n = \frac{504}{6}[/tex]
[tex]^{m + n}C_n = 84[/tex]
Hence, there are 84 possible paths
[tex]^{m + n}C_m[/tex] will also give the same result
