A stereo store is offering a special price on a complete set

A stereo store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component: A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions: (a) In how many ways can one component of each type be selected? (b) In how many ways can components be selected if both the receiver and the compact disc player are to be Sony? (c) In how many ways can components be selected if none is to be Sony? (d) In how many ways can a selection be made if at least one Sony component is to be included? (e) If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component? (Round your answer to three decimal places.) at least one Sony component exactly one Sony component

Solution

Thus,

a) One component of each type can be chosen in :

⁴C₁ * ⁴C₁ * ³C₁ * ⁴C₁ =  4 * 4 * 3 * 4 = 192 ways

b) Since the Receiver and CD player are fixed,

Turntable caqn be chosen in 4 ways and speakers can be chosen in 3 ways.

So total number of ways = 4 * 3 = 12 ways

c)

IF none of them is sony then,

Number of ways to select each are:

= 3 * 3 * 3 * 3

= 81 ways

d) Atleast one sony component

= Total number of ways - no sony component

= 192 - 81

= 111 ways

e)

Probability of atleast 1 sony component = 111 / 192 = 0.5781

Number of ways of exactly selecting one sony component =

P ( Sony Receiver) + P (Sony CD Player) + P (Sony Turtable)

= (3 * 3 * 3) + (3 * 1 * 3 * 3) + (3 * 3 * 3)

= 81 ways

Probability = 81 / 192

= 0.4218

Hope this helps.

Components	No. Of choices
Receiver	4
CD Player	4
Speakers	3
Turn Table	4