Question 213282
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! / (12 - 5)! = (12*11*10*9*8*7*6*5*4*3*2*1)/7!}}}
{{{(12*11*10*9*8*7*6*5*4*3*2*1)/(7*6*5*4*3*2*1)}}}
{{{12*11*10*9*8 = 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! / (11 - 4)!}}}
{{{(11*10*9*8*7*6*5*4*3*2*1) / (7*6*5*4*3*2*1)}}}
{{{11*10*9*8 = 7920}}} answer
Now just put Grandma's picture in all the empty middle spaces