A.
If we take 7 paintings to be hung in 7 spaces side by side, the first space can have any one of the 7 paintings, the second space can have any one of the remaining 6 paintings (as 1 is already hung), the third space can have any one of the remaining 5 paintings (as 2 already hung)...It goes on like this.
So we have
ways to arrange all the paintings from left to write. <em>(in factorial notation it is 7!=5040)</em>
B.
We use combinations rather than permutations because order doesn't matter. If we name the paintings A,B,C,D,E,F, and G, groups of 3 paintings of ABC or ACB are the same. So we evaluate
using the combination formula,
![nCr=\frac{n!}{(n-r)!r!}](https://tex.z-dn.net/?f=nCr%3D%5Cfrac%7Bn%21%7D%7B%28n-r%29%21r%21%7D)
We have,
![7C3=\frac{7!}{(7-3)!*3!} = \frac{7!}{4!*3!} = 35](https://tex.z-dn.net/?f=7C3%3D%5Cfrac%7B7%21%7D%7B%287-3%29%21%2A3%21%7D%20%3D%20%5Cfrac%7B7%21%7D%7B4%21%2A3%21%7D%20%3D%2035)
C.
This is similar to part A in some ways. Any 3 pictures can be arranged in
different ways.
. So, 6 different ways.
ANSWER:
A) 5040 ways
B) 35 different groups
C) 6 ways