Interchangin of letters

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chaitanyareddy: Senior | Next Rank: 100 Posts; Posts: 30; Joined: Fri Feb 27, 2009 12:45 am; Thanked: 1 times; GMAT Score:620

by chaitanyareddy » Sat Aug 21, 2010 9:44 am

Q)My name is AJEET. But my son accidentally types the name by
interchanging a pair of letters in my name. What is the probability that
despite this interchange, the name remains unchanged?

Please explain this.

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Source: Beat The GMAT — Problem Solving | Sat Aug 21, 2010 9:44 am

Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Sat Aug 21, 2010 10:08 am

chaitanyareddy wrote:Q)My name is AJEET. But my son accidentally types the name by
interchanging a pair of letters in my name. What is the probability that
despite this interchange, the name remains unchanged?

Please explain this.

Let's start by jotting down the probability formula - whenever a common formula applies to a GMAT question, start by writing it down on your note board.

Probability = (# of desired outcomes)/(total # of possibilities)

So, we need to figure out how many interchanges will leave your name the same and how many total interchanges are possible.

Well, the only way your name could still be spelled correctly is if your son swapped the two "E"s. So, there's only 1 desired outcome.

There are 5 letters in total, and each interchange involves a pair of letters. Accordingly, we need to figure out how many different pairs there are that could be swapped.

When we want to count the number of subgroups that can be formed out of a larger group, we use the combinations formula:

nCk = n!/k!(n-k)!

In this case, there are 5 total letters and we're choosing 2 of them, so:

5C2 = 5!/2!(5-2)! = 5!/2!3! = 5*4/2*1 = 20/2 = 10

Consequently, there are 10 total possibilities.

Finally, we plug in:

Probability = (# of desired outcomes)/(total # of possibilities) = 1/10

(As an aside, please always include the answer choices and the source of your question.)

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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vijaynaik: Senior | Next Rank: 100 Posts; Posts: 74; Joined: Fri Jun 25, 2010 6:33 pm; Thanked: 3 times

by vijaynaik » Sat Aug 21, 2010 5:14 pm

@Stuart, I misinterpreted the question as it said pair of letters so I thought the letters must be adjacent to each other. Hence i came to the answer 1/4.

AJEET -- AJ, JE, EE, ET

My question to you is how do we figure out that it's talking about any character pair?
Excuse me if it is too easy to comprehend.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Sat Aug 21, 2010 6:56 pm

vijaynaik wrote:@Stuart, I misinterpreted the question as it said pair of letters so I thought the letters must be adjacent to each other. Hence i came to the answer 1/4.

AJEET -- AJ, JE, EE, ET

My question to you is how do we figure out that it's talking about any character pair?
Excuse me if it is too easy to comprehend.

We should always choose the simplest interpretation of a word problem.

This problem cites "a pair of letters"; without further clarification, it could be any pair. If the problem had cited "a consecutive pair of letters", then your interpretation would have been correct.

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vijaynaik: Senior | Next Rank: 100 Posts; Posts: 74; Joined: Fri Jun 25, 2010 6:33 pm; Thanked: 3 times

by vijaynaik » Sat Aug 21, 2010 7:50 pm

Thanks a lot Stuart.

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chaitanyareddy: Senior | Next Rank: 100 Posts; Posts: 30; Joined: Fri Feb 27, 2009 12:45 am; Thanked: 1 times; GMAT Score:620

by chaitanyareddy » Sat Aug 21, 2010 11:02 pm

Hi Stuart , thank s alot for the explanation. This question is from a collection of tough questions which my friend gave me. not having any book name in particular.

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