Pavel has to visit his aunt, who lives exactly eight blocks

[varun289]

Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks east of his current location. If Pavel travels only along streets and does not travel diagonally, the shortest possible route connecting the two points is exactly 14 blocks. How many different 14-block routes may Pavel take to travel the shortest possible distance to his aunt's house?


14!/8!.6!

[Atekihcan]

Any such route has to include 8 blocks in the north and 6 blocks in the east.
The difference between any two such routes lies in the order in which Pavel choose to go to the north and to the east. Since Pavel has 14 blocks to cover, he just need to choose which 8 blocks will be to the north as the rest must be to the east and there will be exactly one possible selection for them.

So, number of such routes = number of ways to select 8 blocks from 14 blocks = 14C8 = 14!/[(8!)*(6!)]

[GMATGuruNY]

Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks east of his current location. If Pavel travels only along streets and does not travel diagonally, the shortest possible route connecting the two points is exactly 14 blocks. How many different 14-block routes may Pavel take to travel the shortest possible distance to his aunt's house?

The number of ways to arrange 5 distinct elements = 5!.

How many different ways can the letters in the word SPEED be arranged?
Here, because the arrangement includes IDENTICAL elements -- the two E's -- the total number of possible arrangements will be LESS than 5!.
The reason is that the arrangement DOESN'T CHANGE when the identical elements swap positions.
Since any arrangement of the two E's does not change the total permutation, we divide by the number of ways to arrange the two E's:
5!/2! = 60.

Another example:
The number of ways to arrange the letters in the word RADAR = 5!/(2!2!) = 30.
Here, we divide by 2! to account for the two A's and by another 2! to account for the two R's.

One more:
The number of ways to arrange the letters in the word MISSISSIPPI = 11!/(4!4!2!).
Here, we divide by 4! to account for the four I's, by another 4! to account for the four S's, and by 2! to account for the two P's.

In the problem above, Pavel must travel exactly 8 blocks north (NNNNNNNN) and exactly 6 blocks east (EEEEEE).
Any arrangement of the letters NNNNNNNNEEEEEE represents a possible route.

The number of ways to arrange NNNNNNNNEEEEEE = 14!/(8!6!).

[Brent@GMATPrepNow]

Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks east of his current location. If Pavel travels only along streets and does not travel diagonally, the shortest possible route connecting the two points is exactly 14 blocks. How many different 14-block routes may Pavel take to travel the shortest possible distance to his aunt's house?


14!/8!.6!

Here's one more approach.

As is already mentioned above, Pavel's route will consist of 14 steps (Norths and Easts).
6 of those steps will be Easts and the rest will be Norths.

So, let's select the 6 steps that will be Easts and let the rest be Norths.

In how many ways can we select 6 of the 14 steps?
Well, since the order in which we select the steps does not matter, we can use combinations.
So, we can select the 6 steps in 14C6 ways.
14C6 = 14!/(8!6!)

IMPORTANT: Notice that I said the order in which we select the steps does not matter. In other words, selecting steps 2, 4, 5, 8, 10, and 13 to be Easts is the same as selecting steps 4, 5, 10, 8, 13 and 2. Given this, we can use combinations.

Cheers,
Brent

[Scott@TargetTestPrep]

Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks east of his current location. If Pavel travels only along streets and does not travel diagonally, the shortest possible route connecting the two points is exactly 14 blocks. How many different 14-block routes may Pavel take to travel the shortest possible distance to his aunt's house?


14!/8!.6!

Solution:

Let N be a “north” block (i.e., when Pavel travels north) and E be an “east” block (i.e., when he travels east). Thus, one path Pavel can go to his aunt’s house is NNNNNNNNEEEEEE and the total number of paths is 14! / (8!6!) (i.e., the number of ways one can arrange 8 N’s and 6 E’s in NNNNNNNNEEEEEE).