Telephone calls

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MURALIDHARGAJULA: Junior | Next Rank: 30 Posts; Posts: 13; Joined: Thu Jul 29, 2010 6:44 pm; Thanked: 1 times

Telephone calls

by MURALIDHARGAJULA » Fri Jun 03, 2011 12:07 am

A company has 8 telephone lines. IF 4 calls have to be made by the company , what is the probability that they are each made on a different line?

A) 35/2048
B) 105/256
C) 105/2048
D) 35/256
E) 105/128

[spoiler]OA: B[/spoiler]

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Source: Beat The GMAT — Problem Solving | Fri Jun 03, 2011 12:07 am

cans: Legendary Member; Posts: 1309; Joined: Mon Apr 04, 2011 5:34 am; Location: India; Thanked: 310 times; Followed by:123 members; GMAT Score:750

by cans » Fri Jun 03, 2011 12:16 am

total possible ways = 8^4 (each call can be made on any of the 8 lines)
if different lines, select 4 lines and arrange them - 8C4 * 4!
thus prob = 8C4 * 4! / 8^4 = 8*7*5*6/(8*8*8*8) = 105/256
IMO B

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Fri Jun 03, 2011 12:24 am

MURALIDHARGAJULA wrote:A company has 8 telephone lines. IF 4 calls have to be made by the company , what is the probability that they are each made on a different line?

Total number of ways to make 4 calls using 8 lines = 8^4

But we have to make calls on different lines.
Hence, we have to select 4 lines out of 8 and distribute the 4 calls in those selected lines. Which can be done in (8C4)*4! ways

Therefore, required probability = (8C4)*4!/(8^4) = 8*7*5*6/8*8*8*8 = 105/256

The correct answer is B.

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ganesh prasath: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Sun Sep 12, 2010 9:09 pm

by ganesh prasath » Fri Jun 03, 2011 5:21 am

"Hence, we have to select 4 lines out of 8 and distribute the 4 calls in those selected lines. Which can be done in (8C4)*4! ways"

I didnt understand this part could u please explain this to me..

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Fri Jun 03, 2011 6:00 am

ganesh prasath wrote:"Hence, we have to select 4 lines out of 8 and distribute the 4 calls in those selected lines. Which can be done in (8C4)*4! ways"

I didnt understand this part could u please explain this to me..

The company has 8 lines and they have to make 4 calls each on different line. Hence they need 4 different lines to do so. Hence, we are selecting 4 lines out of 8 lines in 8C4 ways.

Now we have to assign the 4 calls to these 4 selected lines. Which can be done in 4! ways, because the first call can be made in any of the 4 selected lines. The second call can be made in any of the remaining 3 selected lines. The third call can be made in any of the remaining 2 selected lines. And the fourth and last call can be made in the remaining selected line.

Hence, we can assign the 4 calls to these 4 selected lines in 4*3*2*1 = 4! ways

Hope it helps.

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vikram4689: Legendary Member; Posts: 1325; Joined: Sun Nov 01, 2009 6:24 am; Thanked: 105 times; Followed by:14 members

by vikram4689 » Fri Jun 03, 2011 6:32 am

There is one easier way:

Selection of 4 different telephone lines : 8*7*6*5
Selection of 4 telephone lines (no restriction) : 8*8*8*8

Prob. : (8*7*6*5)/(8*8*8*8) = 105/256

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Fri Jun 03, 2011 7:00 am

MURALIDHARGAJULA wrote:A company has 8 telephone lines. IF 4 calls have to be made by the company , what is the probability that they are each made on a different line?

A) 35/2048
B) 105/256
C) 105/2048
D) 35/256
E) 105/128

[spoiler]OA: B[/spoiler]

Another approach:

The first call is irrelevant: it can be placed on any of the 8 lines.

P(2nd call uses a different line from the 1st) = 7/8.
P(3rd call uses a different line from the 1st and 2nd) = 6/8 = 3/4.
P(4th call uses a different line from the 1st, 2nd and 3rd) = 5/8.

Since we want all of the events above to happen together, we multiply the results:
P(all 4 calls use a different line) = 7/8 * 3/4 * 5/8 = 105/256.

The correct answer is C.

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