Arrangements: Five boys and five girls are arranged in a row

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Clifford

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Five boys and Five girls are arranged in a row. How many possible arrangemnts are there if:
A) there are no restrictions
B) a specific guy must be in the rightmost
C) 5 guys are together
D) 5 guys are on one side
E) the boys and girls must alternate

Here is my work, I was wondering if somebody could look it over and let me know if I have done it right or not, thanks.

A) 10! = 3628800
B) (10 - 1)! = 326850
C) treat 5 guys as one person, and multiply by 5, since they can be in any order
6! * 5 = 3600
D) treat them as one person (6 - 1)! = 120
E) treat is as 5 people, and multiply by 5 since they can be in any order
5! * 5 = 600
 
Re: Arrangements

Clifford said:
Five boys and Five girls are arranged in a row. How many possible arrangemnts are there if:
A) there are no restrictions
B) a specific guy must be in the rightmost
C) 5 guys are together
D) 5 guys are on one side
E) the boys and girls must alternate

Here is my work, I was wondering if somebody could look it over and let me know if I have done it right or not, thanks.

A) 10! YES!
B) (10 - 1)! YES!
C) treat 5 guys as one person,(YES) BUT no: 6!(5!)
D) treat them as one person (6 - 1)! = 120NO! : 2(5!)(5!)
E) treat is as 5 people, and multiply by 5 since they can be in any order
5! * 5 = 600 NO! : 2(5!)(5!)
Click to expand...
 
Re: Arrangements: Five boys and five girls are arranged in a

Hello, Clifford!

I want to clarify what pka said . . .

Five boys and five girls are arranged in a row.
How many possible arrangemnts are there if:
A) there are no restrictions
B) a specific guy must be in the rightmost
C) 5 guys are together
D) 5 guys are on one side
E) the boys and girls must alternate
Click to expand...

C) Yes, treat the 5 guys as one person. .(Duct-tape them together.)
The six "people" can be arranged in \(\displaystyle 6!\) ways.
But the 5 guys can be arranged in \(\displaystyle 5!\) ways.
. . Answer: \(\displaystyle \,(6!)(5!)\:=\:86,400\)


D) Yes, treat them as one person.
The guys can be on either end (2 choices).
They can be arranged in \(\displaystyle 5!\) ways
and the girls can be arranged in \(\displaystyle 5!\)ways.
. . Answer: \(\displaystyle \,2(5!)(5!) \:=\:28,800\)


E) They can be arranged: \(\displaystyle \,BGBGBGBGBG\.\) or \(\displaystyle \,GBGBGBGBGB\) . . . 2 ways.
The boys can be arranged in \(\displaystyle 5!\) ways.
The girls can be arranged in \(\displaystyle 5!\) ways.
. . Answer: \(\displaystyle \,2(5!)(5!) \:=\:28,800\)

 
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