Question

Sally is interested in buying a car. If she has a choice of 3 colors (Red, Green, Blue), 2 body types (Two or four doors) and 3 engine types (Four, six or eight cylinders), calculate how many different models can she choose from
A. 6
B. 8
C. 12
D. 16
E. 18

Answer 1

We write the number of categories of each type of feature of the car and find the total number of models available by multiplying the total number of categories from each feature. We use the fundamental principle of counting to calculate the product of all three numbers.

• Fundamental principle of counting is a rule that is used to find the total number of possible outcomes. If we have $x$ ways to do a work and $y$ ways to another work, then the number of ways to do both work is given by $x \times y$.

Complete step by step answer:
We are given three features of the car to decide upon.
Features that Sally has to choose from are color, body type and engine type of the car.
Let us calculate the number of categories available in each feature.
We are given that there are three colors: Red, Green and Blue
$\therefore$Number of categories under the feature color $= 3$ ……….… (1)
We are given that there are two body types: Two doors and four doors
$\therefore$Number of categories under the feature body type $= 2$ ……….… (2)
We are given that there are three engine types: Four cylinders, six cylinders and eight cylinders
$\therefore$Number of categories under the feature engine type $= 3$ ………… (3)
From equations (1), (2) and (3), we have count of models available in each feature.
So, we can calculate the total number of models available from which Sally can choose using the fundamental principle of counting.
$\Rightarrow$ Total number of models $= 3 \times 2 \times 3$
$\Rightarrow$ Total number of models $= 18$

$\therefore$ Total number of models is 18. Option E is the correct option.

Note:
Alternate method:
We can solve this question using combination method
We know if we have to choose $r$ objects from total $n$objects, then the number of ways to choose objects is given by the formula $^n{C_r}$, where $^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$
If $r = 1$
${ \Rightarrow ^n}{C_1} = \dfrac{{n!}}{{(n - 1)!1!}}$
Use the formula of factorial i.e. $n! = n \times (n - 1)!$in the numerator
${ \Rightarrow ^n}{C_1} = \dfrac{{n \times (n - 1)!}}{{(n - 1)!1!}}$
Cancel the same terms from numerator and denominator and write $1! = 1$
${ \Rightarrow ^n}{C_1} = n$ ………...… (1)
Since, we are given
Number of categories under the feature color $= 3$
$\Rightarrow$ Number of ways to choose a car having one of the colors is given by $^3{C_1}$
Number of categories under the feature body type $= 2$
$\Rightarrow$ Number of ways to choose a car having one of the body types is given by $^2{C_1}$
Number of categories under the feature engine type $= 3$
$\Rightarrow$ Number of ways to choose a car having one of the engine type is given by $^3{C_1}$
So, total number of models available to choose from is given by multiplication of choices from each feature
$\Rightarrow$ Number of models ${ = ^3}{C_1}{ \times ^2}{C_1}{ \times ^3}{C_1}$
Use equation (1) to write the values in the product
$\Rightarrow$ Number of models $= 3 \times 2 \times 3$
$\Rightarrow$ Number of models $= 18$
$\therefore$ Number of models is 18.

$\therefore$ Option E is the correct option.