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Counting question - need expert help

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voodoo_child: Master | Next Rank: 500 Posts; Posts: 405; Joined: Thu Feb 10, 2011 1:44 amFollowed by: 1 member; Thanked: 3 times

by voodoo_child » Mon Jul 02, 2012 5:19 am

In how many ways can the letters of the word LEVEL be arranged so that the first letter is and the last letter is ?

4
6
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24

OA - 6

I see how 3! = 6. That is, fix L and E and then we are left with only 6 possibilities for LEV.

My question is that - why aren't we counting two possibilities of choosing the letters L and E? Essentially, let's consider the following diagram :

1 2 3 4 5

there are two ways one can choose L and E for the first and the last position. Why aren't we counting that?

help?

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Source: Beat The GMAT — Problem Solving | Mon Jul 02, 2012 5:19 am

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Jul 02, 2012 7:47 am

voodoo_child wrote:In how many ways can the letters of the word LEVEL be arranged so that the first letter is and the last letter is ?
4
6
10
12
24

I think you're missing part of the question.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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voodoo_child: Master | Next Rank: 500 Posts; Posts: 405; Joined: Thu Feb 10, 2011 1:44 amFollowed by: 1 member; Thanked: 3 times

by voodoo_child » Mon Jul 02, 2012 12:45 pm

Sorry Bret. There was a copy paste error.

In how many ways can the letters of the word LEVEL be arranged so that the first letter is "L" and the last letter is "E" ?
4
6
10
12
24

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GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Jul 02, 2012 1:38 pm

voodoo_child wrote:Sorry Bret. There was a copy paste error.

In how many ways can the letters of the word LEVEL be arranged so that the first letter is "L" and the last letter is "E" ?
4
6
10
12
24

I see.
Your earlier question was "Why don't we count the 2 ways to select the L for the first letter and the 2 ways to select the E for the last letter?" Well, there aren't really 2 ways to make L the first letter since both L's in LEVEL are identical. The same goes for selecting an E for the last letter.
It doesn't matter which L you choose and which E you choose. Once those 2 letters are in place, we have L _ _ _ E, and we must fill the 3 spaces with the 3 letters E, V, and L.

We can fill 3 spaces with 3 different letters in 3! ways (6 ways).

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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