permutation

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kksekar: Junior | Next Rank: 30 Posts; Posts: 16; Joined: Wed Jan 18, 2012 9:18 pm; Thanked: 1 times

by kksekar » Mon May 07, 2012 8:32 am

of the 6 letters of the word square.
a) In how many of them is r the second letter? _ r _ _ _ _
b) In how many of them are q and e next to each other?

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Source: Beat The GMAT — Problem Solving | Mon May 07, 2012 8:32 am

Bill@VeritasPrep: GMAT Instructor; Posts: 1248; Joined: Thu Mar 29, 2012 2:57 pmFollowed by: 192 members; Location: Everywhere; Thanked: 503 times; GMAT Score:780

by Bill@VeritasPrep » Mon May 07, 2012 8:43 am

a) If we set r in the second spot, there are 5 spots for 5 letters. 5!=120

b) We can put qe next to each other in the following ways:

q e _ _ _ _
_ q e _ _ _
_ _ q e _ _
_ _ _ q e _
_ _ _ _ q e

For each of these, we have 4 spots for 4 letters: 4!=24. 24*5=120. However, for each of the arrangements we identified, we can reverse the order of q and e, so we need to double that total. 120*2=240.

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aneesh.kg: Master | Next Rank: 500 Posts; Posts: 385; Joined: Mon Apr 16, 2012 8:40 amFollowed by: 29 members; Location: Pune, India; Thanked: 186 times

by aneesh.kg » Mon May 07, 2012 10:02 am

We have these alphabets to use:
S,Q,U,A,R,E.

a) R has the be the second letter. The remaining 5 alphabets can be arranged on the remaining places in 5! = 120 ways.

b) Since Q and E have to together, lets first group them like this: (Q, E) and treat it as one single entity.
We have these 5 entities now to be arranged: (Q,E),S,U,A,R
They can be arranged in 5! ways. But, the Q and E can be arranged inside the bracket in 2! ways because the question has not mentioned that Q has to come before E.
Total number of ways = 5!*2! = 120*2 = 240

Aneesh Bangia
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