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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)`.
The value of `sum_(i=0)^(10)P(i,10-i)` is

A

`1024`

B

`512`

C

`256`

D

`128`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `S=sum_(i=0)^(10)P(i,10-i)=P(0,10)+P(1,9)+....+P(10,0)`
`=^(10)C_(0)+^(10)C_(1)+….+^(10)C_(10)=2^(10)=1024`
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