Question

You need to hang seven pictures in a straight line

a.) in how many ways can this be accomplished?

b.) if the picture of your great grandfather must be in the middle, how many ways can this be hung?

Answer 1

I'm just going to answer part A because I don't know part B.
part a.) 7*6*5*4*3*2*1=5040
so there are 5040 different ways this can be accomplished

Answer 2

a.) 5040 ways.

b.) 720 ways

Explanation:

a.)

As we have to arrange the seven pictures in a straight line this means we have the following ways to do it:

There are 7 choices for the first place and now 6 choices for the second place then 5 choices for the third place and so on.

Hence, the total number of ways are:

7×6×5×4×3×2×1=7!

=5040 ways

b.)

Now it is given that the middle place is fixed i.e. it is fixed for grandfather.

Hence, now there are total 6 pictures left that are need to be arranged in the 6 given places.

Hence now the total number of such cases possible are:

6!=6×5×4×3×2×1

=720 ways.