# Thread: Counting Problem

1. ## Counting Problem

Say there is a CD player that holds 5 CDs. There are 5 trays that hold one cd each. I have 100 CD's.

In how many ways can the CD player be loaded if AT MOST two CD's are placed in it?

2. Hello, jzellt!

There is a CD player that holds 5 CDs.
There are 5 trays that hold one CD each. I have 100 CD's.

In how many ways can the CD player be loaded if AT MOST two CD's are placed in it?
I assume that the 100 CD's are distinguishable
. . and that the 5 trays are also distinguishable.

Place one CD into the player.
There are 100 choices for the CD.
There are 5 choices for the tray to place it in.
. . There are: . $100 \times 5 \:=\:500$ ways to load one CD into the player.

Place two CD's into the player.
There are ${100\choose2} \:=\:4950$ ways to choose two CD's.
There are: . $_5P_2 \:=\:20$ ways to place them in the trays.
. . There are: . $4950 \times 20 \:=\:99,000$ ways to load two CD's into the player.

Therefore, there are: . $500 + 99,000 \:=\:{\bf{\color{blue}99,500}}$ ways to load at most two CD's.