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 14block 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 has to visit his aunt, who lives exactly eight blocks north and six blocks
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Pavel's route will consist of 14 steps (Norths and Easts).BTGmoderatorDC wrote: ↑Sat Aug 27, 2022 2:32 amPavel 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 14block 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
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!)
Answer: B
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BTGmoderatorDC wrote: ↑Sat Aug 27, 2022 2:32 amPavel 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 14block 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).
Answer: B
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