## Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks

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### Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks

by BTGmoderatorDC » Sat Aug 27, 2022 2:32 am

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## Global Stats

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?

A. 14
B. (14!)/(8!6!)
C. (22!)/(14!8!)
D. 8!6!
E. 14!8!6!

OA B

Source: Veritas Prep

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### Re: Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks

by [email protected] » Sun Aug 28, 2022 9:24 am
BTGmoderatorDC wrote:
Sat Aug 27, 2022 2:32 am
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?

A. 14
B. (14!)/(8!6!)
C. (22!)/(14!8!)
D. 8!6!
E. 14!8!6!

OA B

Source: Veritas Prep
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!)

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### Re: Pavel has to visit his aunt, who lives exactly eight blocks north and six blocks

by [email protected] » Thu Jan 12, 2023 6:05 pm
BTGmoderatorDC wrote:
Sat Aug 27, 2022 2:32 am
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?

A. 14
B. (14!)/(8!6!)
C. (22!)/(14!8!)
D. 8!6!
E. 14!8!6!

OA B

Source: Veritas Prep

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).