I am unable to understand why 17,640 instead of 92,378. I plotted the points on the xy-coordinate plane and calculated that she would have to go 19 times in total. Like she would go to the grocery first from (0,0) for that she could go 4 times east and after reaching (4,0) she could go 6 times North and she would reach at the grocery store. From there, she could go 6 times east and then 3 times north. No matter what ay I chose, east is being repeated 10 times and north is being repeated 9 times. So, if we solve this factorial-
19!/(10!*9!) it is coming out to be 92,378.
So, I want to understand where am I going wrong and what is a faster way to solve this problem and other such direction problems as I spent 10-15 mins on this.
You jumped a step too far. The reason you cannot simply do 19C9 (i.e, \frac{19!}{10! 9!}) is that that is akin to skipping the grocery store constraint completely. One routing that would violate that rule is 10 consecutive Ns and 9 consecutive Es - you never cross the grocery store there.
Yeah, you are right! But with just 6 Ns she can’t reach home and the time I cross 6 Ns there is a risk of skipping grocery store. So, that means order would matter in this problem right. She can’t go south otherwise this could have been easily solved. Could you give some more hint as to what other approach I can try or could we subtract such cases where she could skip the store?
