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 beselected? b. In how many ways can components be selected if boththe receiver and the compact disc player are to be Sony? c. In how many ways can components be selected if none isto be Sony? d. In how many ways can a selection be made if at least oneSony component is to be included? e. If someone flips switches on the selection in a completelyrandom fashion, what is the probability that thesystem selected contains at least one Sony component? Exactly one Sony component?

Question:

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

components (receiver, compact disc player, speakers, cassette

deck). A purchaser is offered a choice of manufacturer for each

component:

Receiver: Kenwood, Sony, Sherwood

Compact disc player: Onkyo, Pioneer, Sony, Technics

Speakers: Boston, Infinity, Polk

Turn table: Onkyo, Sony, Teac, Technics

Answer:

The answers to the questions are;

(a) 144

(b) 12

(c) 54

(d) 90

(e) 0.625, 0.4375

Step-by-step explanation:

We apply the multiplication rule to solve questions (a) to (d) as follows

a. In how many ways can one component of each type beselected?

Receiver = 3 ways

Compact disc = 4 ways

Speaker = 3 ways and

Turn table = 4 ways

Therefore we have

One component of each can be selected in

3 * 4 * 3 * 4 = 144 ways

Answer = 144 ways

b. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?

If the receiver MUST be Sony then that is one way out of 4

Similarly too for the compact disc player.

Therefore we have the following ways

1*1*3*4 = 12

Answer = 12 ways

c. In how many ways can components be selected if none is to be Sony?

If NONE is to be Sony, we then have to exclude Sony in our calculations as follows

Receiver becomes 2 ways

Compact disc is then 3 ways

Speaker remains the same 3 because no Sony there and

Turn table becomes 3

Therefore we have

2*3*3*3 = 54

Answer = 54 ways

d. In how many ways can a selection be made if at least one Sony component is to be included?

Here, we use the logic that if at least one is to be Sony then

(The number of ways of selecting one component) - (The number of ways where none is Sony)

= 144 - 54 = 90

Answer = 90 ways

e. If someone flips switches on the selection in a completelyrandom fashion, what is the probability that thesystem selected contains at least one Sony component?

The probability is (Number of required outcomes) : (Number of possible outcomes) = 90/144 = 0.625

Answer = 0.625

Exactly one Sony component?

Here again we have

If the receiver MUST be Sony then

1*3*3*3 = 27

If the compact disc MUST be Sony we have after removing Sony from receiver and turn table

2*1*3*3 = 18

Similarly, if the turn table MUST be Sony, then after removing Sony from the receiver and compact disc

2*3*3=18

Total number of ways = 63

Probability = 63/144 = 0.4375