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?
Hello, jzellt!
I assume that the 100 CD's are distinguishableThere 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?
. . 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: .$\displaystyle 100 \times 5 \:=\:500$ ways to load one CD into the player.
Place two CD's into the player.
There are $\displaystyle {100\choose2} \:=\:4950$ ways to choose two CD's.
There are: .$\displaystyle _5P_2 \:=\:20$ ways to place them in the trays.
. . There are: .$\displaystyle 4950 \times 20 \:=\:99,000$ ways to load two CD's into the player.
Therefore, there are: .$\displaystyle 500 + 99,000 \:=\:{\bf{\color{blue}99,500}}$ ways to load at most two CD's.