Question

Consider a city whose streets are laid out as a grid. Streets run north-south or east-west. A visitor is dropped off by a cab at a certain intersection and would like to visit a museum that is 6 blocks to the east and 3 blocks to the north of his current location. The visitor always takes a direct path and never travels west or south in the course of his walk. How many distinct paths are there for the visitor to walk to the museum

Answer 1

Answer:

84 possible paths

Step-by-step explanation:

Given

$N =3$ --- 3 blocks north

$E = 6$ --- 6 blocks east

Required

Number of distinct path

To solve this question, we make use of the following formula

$^{m+n}C_n \ =\ ^{m+n}C_m$

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:

$m = 3$

$n = 6$

So, we have:

$^{m + n}C_n = ^{3+6}C_6$

$^{m + n}C_n = ^9C_6$

Apply combination formula

$^{m + n}C_n = \frac{9!}{(9-6)!6!}$

$^{m + n}C_n = \frac{9!}{3!*6!}$

Expand the numerator

$^{m + n}C_n = \frac{9*8*7*6!}{3!*6!}$

$^{m + n}C_n = \frac{9*8*7}{3!}$

Expand the denominator

$^{m + n}C_n = \frac{9*8*7}{3*2*1}$

$^{m + n}C_n = \frac{504}{6}$

$^{m + n}C_n = 84$

Hence, there are 84 possible paths

$^{m + n}C_m$ will also give the same result