SOLUTION: A music class of 5 girls and 4 boys is having a recital. If each member is to perform once,how many ways can the program be arranged in each of the following cases? (A) all gir

It cannot be combinations, because these are ORDERINGS of performances
Permutations are ORDERINGS of choices.  Combinations are CHOICES only.
So if it were combinations, then the answer would be 1, which means that
all 9 people are chosen to perform, regardless of what order they perform
in.  Since the ORDER in which they perform is the only thing that matters
in any of these problems, these are PERMUTATIONS, not combinations.
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Choose one of 5 girls to perform 1st.  
That's 5 ways to choose the first performer.

Choose one of the 4 remaining girls to perform 2nd.
That's 5×4 ways to choose the order of the first 2 performers. 

Choose one of the 3 remaining girls to perform 3rd.
That's 5×4×3 ways to choose the order of the first 3 performers.

Choose one of the 2 remaining girls to perform 4th.
That's 5×4×3×2 ways to choose the order of the first 4 performers.

Choose the 1 remaining girl to perform 5th.
That's 5×4×3×2×1 ways to choose the order of the first 5 performers.

So far that 5! or 5×4×3×2×1 = 120 = P(5,5) permutations 
or WAYS TO ORDER the 5 girls, taking all 5 of them.

Choose one of 4 boys to perform 6th.  
That's 5×4×3×2×1×4 ways to choose the order of the first 6 performers.

Choose one of the 3 remaining boys to perform 7th.
That's 5×4×3×2×1×4×3 ways to choose the order of the first 7 performers. 

Choose one of the 2 remaining boys to perform 8th.
That's 5×4×3×2×1×4×3×2 ways to choose the order of the first 8 performers. 

Choose the 1 remaining boy to perform 9th or last.
That's 5×4×3×2×1×4×3×2×1 ways to choose the order of all 9 performers. 

The short way to do this is P(5,5)×P(4,4) = (5×4×3×2×1)×(4×3×2×1) = 
120×24 = 2800. 
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Choose one of 5 girls to perform 1st.  
That's 5 ways to choose the first performer.

Choose one of 4 boys to perform last or 9th.  
That's 5×4 ways to choose the 1st and 9th performers.

Choose one of the 7 remaining people to perform 2nd.
That's 5×4×7 ways to choose the order of the 1st 2 and 9th performers. 

Choose one of the 6 remaining people to perform 3rd.
That's 5×4×7×6 ways to choose the order of the 1st 3 and 9th performers.

Choose one of the 5 remaining people to perform 4th.
That's 5×4×7×6×5 ways to choose the order of the 1st 4 and 9th performers.

Choose one of the 4 remaining people to perform 5th.
That's 5×4×7×6×5×4 ways to choose the order of the 1st 5 and 9th performers.

Choose one of the 3 remaining people to perform 6th.
That's 5×4×7×6×5×4×3 ways to choose the order of the 1st 6 and 9th performers.

Choose one of the 2 remaining people to perform 7th.
That's 5×4×7×6×5×4×3×2 ways to choose the order of the 1st 7 and 9th performers.

Choose the 1 remaining person to perform 8th.
That's 5×4×7×6×5×4×3×2×1 ways to choose the order of the 1st 8 and 9th performers.  

So the answer is:  5×4×7×6×5×4×3×2×1 = 100800.

Short way:  Choose the 1st girl P(5,1), Choose the last boy P(4,1),
Choose the orderings for the middle 7 P(7,7)

Answer: P(5,1)×P(4,1)×P(7,7) = 5×4×5040 = 100800  
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Choose Elsa P(1,1)=1 way.  Choose Doug P(1,1) = 1 way.  Choose the
middle 7 performers P(7,7) ways.  

Answer: P(1,1)×P(1,1)×P(7,7) = 1×1×5040 = 5040 orderings.

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Since there are 5 girls the program must go GBGBGBGBG.  We choose
the odd numbered performers as the girls in P(5,5) ways, and for
each of those orderings for the girls, we choose
the odd numbered performers as the boys in P(4,4) ways.

That's P(5,5)×P(4,4) = 120×24 = 2880 orderings. 
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Choose the 1st, 5th, and 9th performers as P(5,3). Choose the
orderings for the other performers from the 6 remaining people P(6,6).  

Answer P(5,3)×P(6,6) = (5×4×3)

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Edwin