Answer:
A. 362,880
B. 4,320
C. 14,400
D. 21,600
Step-by-step explanation:
A.
If there are no restrictions as to how they are seated, then we have permutations of 9 elements and there are 9! (factorial of 9) = 362,880 different seating arrangements.
B.
If the boys sit in the middle three seats, they can sit in 3!=6 different ways, the girls can sit then in 6!=720 different ways. By the fundamental rule of counting, there are 6*720 = 4,320 different seating arrangements.
C.
We now have arrangements of the type
g, g, x, x, x, x, x, x, g
The three girls at the ends can be chosen in C(6;3) (combinations of 6 taken 3 at a time) =
different ways. The 6 in the middle can be sit in 6!=720 different ways.
By the fundamental rule of counting, there are 20*720 = 14,400 different seating arrangements.
D.
Now we have arrangements
g,b,g,b,x,x,x,x,x
For the 1st position we have 6 possibilities, for the 2nd we have 3 possibilities, for the 3rd we have 5 possibilities and for the 4th we have 2 possibilities. For the last 5 we have 5!=120 possibilities.
By the fundamental rule of counting, there are 6*3*5*2*120 = 21,600 different seating arrangements.