My Math Forum Permutations

 February 20th, 2012, 09:29 AM #1 Newbie   Joined: Feb 2012 Posts: 5 Thanks: 0 Permutations How many three letter arrangements are there of the letters taken from the word SILLY? (Hint: Consider cases with and without two L's)
February 20th, 2012, 12:09 PM   #2
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Joined: Dec 2006
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Re: Permutations

Hello, koolkidx45!

Quote:
 How many three-letter arrangements are there of the letters taken from the word SILLY?

There are 3 cases to consider:

No L's
[color=beige]. . [/color]We have: $\{S,\,I,\,Y\}$
[color=beige]. . [/color]$\text{There are: }\,3!\,=\,6\text{ arrangements with no L.}$

One L
[color=beige]. . [/color]We have: $L\text{ and }\{S,\,I,\,Y\}$
[color=beige]. . [/color]$\text{There are: }\,{3\choose2} \,=\,3\text{ ways to choose the other 2 letters.}$
[color=beige]. . [/color]$\text{The three letters can be arranged in }3!\,=\,6\text{ ways.}$
$\text{There are: }\:3\,\cdot\,6\:=\:18\text{ arrangements with one L.}$

Two L's
[color=beige]. . [/color]We have $L,\,L\text{ and }\{S,\,I,\,Y\}.$
[color=beige]. . [/color]There are $3\text{ choices for the third letter.}{$
[color=beige]. . [/color]$\text{The three letters can be arranged in }\,{3\choose2}\,=\,3\text{ ways.}$
$\text{There are: }\:3\,\cdot\,3\:=\:9\text{ arrangements with two L#39;s}$

$\text{Therefore, there are: }\:6\,+\,18\,+\,9 \:=\:33\text{ three-letter arrangments.}$

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