# The owner of a stereo store wants to advertise that he has many different sound systems in stock. The store carries 7 different CD players, 8 different receivers and 10 different speakers. How many different sound systems can the owner advertise?

Jun 12, 2018

The owner can advertise a total of 560 different sound systems!

#### Explanation:

1 Speaker (system), 1 Receiver, 1 CD Player

If we only had 1 option for speakers and CD players, but we still have 8 different receivers, then there would be 8 combinations. If we only fixed the speakers (pretend that there is only one speaker system available), then we can work down from there:

$S , {R}_{1} , {C}_{1}$
$S , {R}_{1} , {C}_{2}$
$S , {R}_{1} , {C}_{3}$
...
$S , {R}_{1} , {C}_{8}$
$S , {R}_{2} , {C}_{1}$
...
$S , {R}_{7} , {C}_{8}$

I'm not going to write every combination, but the point is that even if the number of speakers is fixed, there would be:

${N}_{\text{Receiver"xxN_"CD Player}}$

$7 \times 8 = 56$

different combinations! Now, we'll add ANOTHER layer of complexity by considering the options for speakers:

${C}_{\text{Total"=N_"Speaker"xxN_"Receiver"xxN_"CD Player}}$

${C}_{\text{Total}} = 10 \times 7 \times 8$

color(green)(C_"Total"=560