SOLUTION: it would be greatly appreciated if someone helped with the steps on this one.. the answer for a) is 95 040 and the answer for b) is 7920 Twelve different portraits are in the L

Algebra ->  Permutations -> SOLUTION: it would be greatly appreciated if someone helped with the steps on this one.. the answer for a) is 95 040 and the answer for b) is 7920 Twelve different portraits are in the L      Log On


   

Question 213282: it would be greatly appreciated if someone helped with the steps on this one..
the answer for a) is 95 040 and the answer for b) is 7920
Twelve different portraits are in the Ling family's collection
a) In how many ways could five of them be hung in a row on the living room wall?
b)If Grandma Ling's picture must be included and must be hung in the middle of the group of five, how many different arrangements are there?
Found 2 solutions by checkley77, josmiceli:
Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
12*11*10*9*8=95,040 ans.
11*10*1*9*8=7,920 ans.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
a)
when the problem says "How many ways" it
is implying that order makes a difference
since the order you see is a different
"way" to hang them.
When order makes a difference, what you want
is the Permutation of n objects taken k
at a time.
In this problem it is 12 objects taken 5 at a time
which equals 12%21+%2F+%2812+-+5%29%21+=+%2812%2A11%2A10%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%2F7%21
%2812%2A11%2A10%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%287%2A6%2A5%2A4%2A3%2A2%2A1%29
12%2A11%2A10%2A9%2A8+=+95040
b)
Now the problem wants to treat 1 picture diffently from the rest
It will always be in the middle of the group of 5
Imagine that you have thousands of rooms. Now imagine that you hang all the
possible arrangements of the other 11 pictures in groups of
4 in each of the rooms (You've got a huge warehouse full of pictures)
Imagine that you leave a space in the middle of each group of
4 to put Grandma's picture in later.
the number of rooms you need for a different arrangement
in each room is:
11%21+%2F+%2811+-+4%29%21
%2811%2A10%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29+%2F+%287%2A6%2A5%2A4%2A3%2A2%2A1%29
11%2A10%2A9%2A8+=+7920 answer
Now just put Grandma's picture in all the empty middle spaces