Question

If sum of the ordinate and abscissa of a point $P(x,y)$ is $2n(x,y$ are natural number), then the probability that $P$ does not lie on the line $y=x$ is

• A. $\frac{n-1}{n+3}$
• B. $\frac{{}^{2n}{C}_n}{2^{2n}}$
• C. $\frac{2n-2}{2n-1}$
• D. $\frac{2n+1}{2n+3}$

Answer 1

Answer: (C)

$\frac{2n-2}{2n-1}$

Given ordinate + abscissa = 2n

$x+y=2n...\left(1\right)$

Given x, y are natural

$x\ne y\ne 0.$

If x = y

$x+x=2n$

$2x=2n$

$x=n$ as $x=y$ So $y=n$

No possible of eq (1) (x,y)

$(1,2n-1);(2,2n-2),(3,2n-3)...(2n-1,1)$

Total no. of possible combination $\Rightarrow 1$ to $2n-1$

$=2n-1$

Total no. of possible combination p (x,y) lie on

$x=y$ is 1

[As per given p does not lie on x]

Total no. of combination where p does not lie on

x =y line $\Rightarrow 2n-1-1=2n-2$

probability p does not lie on $x=y=\frac{2n-2}{2n-1}.$