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

Answer»

`1024`
`512`
`256`
`128`

SOLUTION :`(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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