Question 1206384
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Answer:  <font color=red size=4>120</font>


Explanation


A,B,C = Ronnie, Annie, Rachel
D,E,F = the other 3 people


A must be to the left of B.
C must be to the right of B.
In other words: we must have A__B__C where 0 or more letters will go in the blanks.


Imagine that persons D,E, and F will select numbers 1 through 6 from a hat. The selections are done without replacement. The numbers refer to seating arrangement.
1 = left most seat
6 = right most seat
Use the nCr combination formula (or Pascals Triangle) to determine there are p = 6C3 = 20 different ways to do this where order doesn't matter.
Eg: Group {1,3,4} is the same as {3,1,4}


Once persons D,E, and F find their seat, there are q = 3! = 3*2*1 = 6 ways to permute the 3 members of that group.
Think of it like a game of musical chairs except none of the chairs get taken away and everyone is able to get a seat.


Therefore, we have p*q = 20*6 = <font color=red>120</font> ways to ensure that the ordering is A,B,C where there may or may not be a gap between Ronnie, Annie, Rachel.


A few examples of valid arrangements:
ABC,DEF
ABD,CEF
ADB,ECF


Examples that aren't valid arrangements
CAB,DEF
CAD,FEB
Both aren't valid because C is not to the right of B.
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