Saved Bookmarks
| 1. |
A path of length n is a sequence of points (x_(1),y_(1)), (x_(2),y_(2)),….,(x_(n),y_(n)) with integer coordinates such that for all i between 1 and n-1 both inclusive, either x_(i+1)=x_(i)+1and y_(i+1)=y_(i) (in which case we say the i^(th) step is rightward) or x_(i+1)=x_(i) and y_(i+1)=y_(i)+1 ( in which case we say that the i^(th) step is upward ). This path is said to start at (x_(1),y_(1)) and end at (x_(n),y_(n)). Let P(a,b), for a and b non-negative integers, denotes the number of paths that start at (0,0) and end at (a,b). Number of ordered pairs (i,j) where i ne j for which P(i,100-i)=P(i,100-j) is |
|
Answer» `50` and `P(j,100-j)=^(100)C_(j)` Given `'^(100)C_(i)=^(100)C_(j)` `impliesi+j=100` NUMBER of non negative integral solutions for this EQUATION `=101` (including `i=j=50`) HENCE required number of ORDERED pairs `=100` |
|