probablity qn

kishokbabu wrote:John and Bill are among 5 runners in a race where there are no ties. How many ways is it possible for john to finish the race ahead of Bill?

A) 5
B) 10
C) 30
D) 60
E) 120

There are 5! (120) number of ways a race with 5 runners could end

There are two kinds of cases
Either John finishes after Bill or John finishes b4 Bill

The number of way John finishes ahead of Bill is exactly half of total possibilities => 120/2 => 60

Option D

Option C

60 ways

If John comes 1st, then Bill should be among 2 to 5 position, so ways = 4
If John comes 2nd, then Bill should be among 3 to 5 position, so ways = 3
If John comes 3rd, then Bill should be among 4 to 5 position, so ways = 2
If John comes 4th, then Bill should be 5th, so ways = 1

Total = 4 + 3+ 2+ 1 = 10
At the same time, other 3 racer's positions can be adjusted in 3! ways.

So 3! . 10 = 60
Option: D

or vice verse-
if B is the last one, then J has 4 options
if B is in the 4th place, then J has 3 options
if B is in the 3rd place, then J has 2 options
if B is in the 2nd place, then J has only 1 option
so, (4+3+2+1)*3!=60 (3!-the way to arrange the rest 3 ppl)