Two children want to walk together to visit their aunt, who lives exactly 7 blocks north and 5 blocks east of their current location. If they travel only along streets and do not travel diagonally, the shortest possible route connecting the two points is exactly 12 blocks. How many different 12-block routes may they take to travel the shortest possible distance to their aunt’s house? Choose the correct answer.
A) 12!/(7*5)  B) 19!/(12!)  C) 12!/(7!5!)  D) 12!*7!*5!  E) 7!*5!
If they need to walk 7 blocks north and 5 blocks east, we can express this route as NNNNNNNEEEEE. The question thus asks us how many ways we can arrange those letters.   The number of arrangements of 12 things is 12!   We then divide by the repetitions of N's and E's, which are 7! and 5! respectively.   The number of walking routes is 12!/(7!5!).   The answer is 12!/(7!5!).
<result>C</result>
Two children want to walk together to visit their aunt, who lives exactly 7 blocks north and 5 blocks east of their current location. If they travel only along streets and do not travel diagonally, the shortest possible route connecting the two points is exactly 12 blocks. How many different 12-block routes may they take to travel the shortest possible distance to their aunt’s house?