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[thegmatbeater]

John and Bill ara among 5 runners in arace 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

[sudhir3127]

IMO D

[Chitts]

Is the anwser 10. I can explain if its correct

[dbart06]

imo = 60

[Chitts]

if John finishes 1st bill can come 2nd,3rd, 4th, 5th. -- 4 ways
if john finishes 2nd bill can come 3rd, 4th, 5th -- 3 ways
if John finishes 3rd bill can come 4th, 5th -- 2
if John finishes 4th bill can come 5th -- 1

Total is 4+3+2+1 = 10..
Is this wrong?

[sudhir3127]

i used a very simple approach ...

As there are 5 people there are 5! ways to arrange them
ie 120 ways to arrange them ...

So, in half of them one guy has to be ahead of the other and vice versa

that gives us 60 as answer..

[acecoolan]

if John finishes 1st bill can come 2nd,3rd, 4th, 5th. -- 4 ways
if john finishes 2nd bill can come 3rd, 4th, 5th -- 3 ways
if John finishes 3rd bill can come 4th, 5th -- 2
if John finishes 4th bill can come 5th -- 1

Total is 4+3+2+1 = 10..
Is this wrong?

Chitts ..you are just looking at the ways in which u can arrange these 2 persons ..u also need to look at the other peple. So add that ur logic

John is 1st - the remaining 4 people have 4! = 24 ways
John is 2nd - bill has 3 positions and the other people have 3! - hence 3! * 3 = 18
John is 3rd - bill has 2 postns and others have 3!, hence 3! * 2 = 12
John is 4th - bill has 1 postn and others 3!, hence 3! * 1 = 6

Add all 24 + 18 + 12 + 6 = 60

But Sudhir's approach is faster and a more intelligent way ..

[pepeprepa]

The problem is: How many ways
Either you understand ways as the number of possibilities of the whole runners
Or you understand as the different possibilities of ranking of the two guys (1st and 3rd / 1st and 2nd .....)

[AleksandrM]

5!/2!(5 - 2)! = 10

John has to come in ahead of Bill. The places are:

5 4 3 2 1

This means that John can come in ahead of Bill in the following ways.

5 4

5 3

5 2

5 1

4 3

4 2

4 1

3 2

3 1

2 1

Add them up, and you get 10 ways.

[acecoolan]

So what is the OA for this ?

I agree pepeprepa - so am not sure how to intepret the question then