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A path of length n is a sequence of poin...

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)+1 ` and `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(j,100-j)` is

A

(a) `50`

B

(b) `99`

C

(c) `100`

D

(d) `101`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` `P(i,100-i)=^(100)C_(i)`
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`
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