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 sum P(43,4)+sum_(j=1)^(5)P(49-j,3) is equal to

Answer»

`P(4,48)`
`P(3,49)`
`P(4,47)`
`P(5,47)`

Solution :`(a)` `P(i,N-i)=^(n)C_(i)=P(n-i,i)`
Now `P(43,4)+sum_(j=1)^(5)P(49-j,3)`
`=('^(47)C_(4)+^(47)C_(3))+^(48)C_(3)+^(49)C_(3)+^(50)C_(3)+^(51)C_(3)`
`=('^(48)C_(4)+^(48)C_(3))+^(49)C_(3)+^(50)C_(3)+^(51)C_(3)`
`=('^(48)C_(4)+^(49)C_(3))+^(50)C_(3)+^(51)C_(3)=('^(50)C_(4)+^(50)C_(3))`
`=^(51)C_(4)+^(51)C_(3)=^(52)C_(4)=P(48,4)=P(4,48)`


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