Home
Class 12
MATHS
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 sum `P(43,4)+sum_(j=1)^(5)P(49-j,3)` is equal to

A

`P(4,48)`

B

`P(3,49)`

C

`P(4,47)`

D

`P(5,47)`

Text Solution

Verified by Experts

The correct Answer is:
A
`(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)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE|Exercise Multiple Correct Answer|4 Videos

  • AREA UNDER CURVES

    CENGAGE|Exercise Question Bank|10 Videos

  • CIRCLE

    CENGAGE|Exercise MATRIX MATCH TYPE|6 Videos

Similar Questions

Explore conceptually related problems

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 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(i,100-j) is

The value of sum_(i=1)^(13) (n^(n) + i^(n-1)) is

Find the distance between P( x_(1), y_(1) ) and Q( x_(2), y_(2) ) when : i. PQ is parallel to the y-axis, ii. PQ is parallel to the x-axis.

If ((1+i)/(1-i))^(x)=1 , then

Let A(x_(1),y_(1),z_(1)),B(x_(2),y_(2),z_(2)) and C(x_(3),y_(3),z_(3)) be the vertices of DeltaABC . i. Find the midpoint D of BC. ii. Find the coordinates of the centroid of DeltaABC .

If A=[(1,2,2),(2,1,-2),(x,2,y)] is a matrix such that "AA"^(T) =9I , find the values of x and y.

Find the values of the real numbers x and y, if the complex numbers (3 - i) x - (2-i) y + 2i + 5 and 2x + (-1 + 2i)y + 3 + 2i are equal