varun289 wrote: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!).
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