Bunuel wrote:
Gerald and his family are at point A when they receive an alert that a cloud of poisonous gas is moving toward them from the southwest. They need to avoid the cloud of gas by driving to a shelter at point B. If they must stay on the grid and are only able to drive north and east, how many different routes can they take from point A to point B?
A. 12
B. 15
C. 21
D. 30
E. 35
Key point: Regardless of how Gerald gets from point A to point B, his path will always consist of driving three blocks north and four blocks east.
For example, one possible route might be represented by: NNNEEEE
Here Gerald drives north for a block, north for another block, north for another block, and then east for four blocks.
Another possible route might be: NEENNEE, etc
So the question really comes down to " in how many different ways can we arrange three N's and 4 E's?"
----ASIDE-------------------
When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:
If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....] So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:
There are
11 letters in total
There are
4 identical I's
There are
4 identical S's
There are
2 identical P's
So, the total number of possible arrangements =
11!/[(
4!)(
4!)(
2!)]
--------------------------
Back to the question.
We want to arrange the letters in the word NNNEEEE
There are
7 letters in total
There are
4 identical E's
There are
3 identical N's
So, the total number of possible arrangements =
7!/[(
4!)(
3!)]
= (7)(6)(5)
(4)(3)(2)(1)/
(4)(3)(2)(1)(3)(2)(1)
= (7)(
6)(5)/(3)(2)(1)
= (7)(5)
= 35
Answer: E
Cheers,
Brent