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# Different seating arrangement

Posted on 2011-10-28
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The school board consists of three men and four women

a. When they hold a meeting, they sit in a row. How many different seating arrangements are there ?  Ans. 5,040

b. How many different ways can the row be arranged if no two women sit next to each other? Ans. 144

Can any one please explain how to get these numbers ? Thanks.
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Question by:mustish1
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Expert Comment

ID: 37047140
a - 7!  =  5040,    7 people to chose in first seat , 6 in the next ,5 in the next  then 4, then 3, then 2 and finally only 1 person left for the last seat

b- are you sure that 144 is correct?
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Author Comment

ID: 37047154
Yes this is 144. Can you please tell me its a permutation or combination?
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Accepted Solution

sdstuber earned 2000 total points
ID: 37047155
for b I get 1440  possible arrangements.

first,  lets look at just Male/Female arrangements,  there are only 10 of those that are legal.

mfmmfmf
mmfmfmf
fmfmmfm
fmmfmfm
fmfmmmf
mfmfmfm
mfmfmmf
fmmmfmf
fmmfmmf
fmfmfmm

so, now it's simply a matter of how many ways can you arrange the men and women within each of those 10,  THAT is 144.

4! * 3!  (24 ways to arrange 4 men and 6 ways to arrange 3 women)

so,  144 * 10 = 1440 total possible arrangements

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Author Comment

ID: 37047168
may be 144 is wrong
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LVL 74

Expert Comment

ID: 37047183
oops,  I had the numbers backwards,  I read it as 4 men, 3 women,  not 3 men , 4 women

for 3men, 4 women  there is only one arrangement

fmfmfmf

so, 4! x 3! x 1 = 144

sorry for the confusion
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Author Comment

ID: 37047194
it shows 4 man and 3 woman
mfmmfmf
mmfmfmf
fmfmmfm
fmmfmfm
fmfmmmf
mfmfmfm
mfmfmmf
fmmmfmf
fmmfmmf
fmfmfmm

but in the question it says 3 man and 4 woman
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LVL 74

Expert Comment

ID: 37047201
yes,  I corrected that in the previous post
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Author Comment

ID: 37047209
CAN YOU PLEASE EXPLAIN THAT PART

How many different ways can the row be arranged if no two women sit next to each other?

for 3men, 4 women  there is only one arrangement

fmfmfmf

so, 4! x 3! x 1 = 144
0

LVL 74

Expert Comment

ID: 37047230
what do you need explained?

1 - there is only one arrangement,  that should hopefully be obvious.  I used no math for that, I simply listed possibilities until I ran out, and I ran out of options after 1.

4! - this is the number of ways the women can be arranged in the row (we're ignoring the men here)  - this is the same logic as the 7! in part A

3! - this is the number of ways the men can be arranged in the row (we're ignoring the women here) - again, same logic as the 7! in part A

we multiply them together because for each arrangement of  women you can have each arrangement of men - their individual seating is not dependent on the other.
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