Permutations and Combinations

You have 6 pictures A, B, C, D, E and F. However, you have only 4 frames (side by side on the wall) where you can place just 4 pictures - one in each frame.

a) In how many ways can you arrange 4 pictures in the 4 frames on the wall?

You have 6 pictures to choose from for the first frame.

For each of those 6 you have 5 left to choose from for the second frame.

4 to choose from for the third frame.

3 to choose from for the fourth frame.

[spoiler]6 x 5 x 4 x 3 = 360[/spoiler] different ways to arrange 4 of 6 pictures

b) In how many ways can you arrange 4 pictures, if every arrangement must have the picture A in it?

There are 4 different places in which to put picture A.

For each of those 4 places, you have three frames left.

You have 5 choices for the first frame, leaving 4 choices for the second frame and 3 choices for the third frame.

[spoiler]4 x 5 x 4 x 3 = 240[/spoiler] different arrangements

c) Your sister too wanted to arrange 4 pictures in the 4 frames, but she would never include the picture A in any of the arrangements. In how many ways can she do that?

In this case you have only 5 pictures to arrange in the 4 frames.

5 choices for the first frame.

For each of those 5, there are 4 choices for the second frame.

3 choices for the third frame.

2 choices for the fourth frame.

[spoiler]5 x 4 x 3 x 2 = 120[/spoiler] different arrangements

For background on how this all works, see this post.

https://infinitemindprep.com/permutation ... difficult/