# 13.5 Counting principles  (Page 6/12)

 Page 6 / 12

## Verbal

For the following exercises, assume that there are $n$ ways an event $A$ can happen, $m$ ways an event $B$ can happen, and that are non-overlapping.

Use the Addition Principle of counting to explain how many ways event can occur.

There are $\text{\hspace{0.17em}}m+n\text{\hspace{0.17em}}$ ways for either event $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ or event $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ to occur.

Use the Multiplication Principle of counting to explain how many ways event can occur.

Answer the following questions.

When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?

The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.

Describe how the permutation of $n$ objects differs from the permutation of choosing $r$ objects from a set of $n$ objects. Include how each is calculated.

What is the term for the arrangement that selects $r$ objects from a set of $n$ objects when the order of the $r$ objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?

A combination; $\text{\hspace{0.17em}}C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}\text{\hspace{0.17em}}$

## Numeric

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

Let the set $A=\left\{-5,-3,-1,2,3,4,5,6\right\}.$ How many ways are there to choose a negative or an even number from $\mathrm{A?}$

Let the set $B=\left\{-23,-16,-7,-2,20,36,48,72\right\}.$ How many ways are there to choose a positive or an odd number from $A?$

$\text{\hspace{0.17em}}4+2=6\text{\hspace{0.17em}}$

How many ways are there to pick a red ace or a club from a standard card playing deck?

How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?

$\text{\hspace{0.17em}}5+4+7=16\text{\hspace{0.17em}}$

How many outcomes are possible from tossing a pair of coins?

How many outcomes are possible from tossing a coin and rolling a 6-sided die?

$\text{\hspace{0.17em}}2×6=12\text{\hspace{0.17em}}$

How many two-letter strings—the first letter from $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and the second letter from $\text{\hspace{0.17em}}B—$ can be formed from the sets $\text{\hspace{0.17em}}A=\left\{b,c,d\right\}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B=\left\{a,e,i,o,u\right\}?\text{\hspace{0.17em}}$

How many ways are there to construct a string of 3 digits if numbers can be repeated?

$\text{\hspace{0.17em}}{10}^{3}=1000\text{\hspace{0.17em}}$

How many ways are there to construct a string of 3 digits if numbers cannot be repeated?

For the following exercises, compute the value of the expression.

$\text{\hspace{0.17em}}P\left(5,2\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}P\left(5,2\right)=20\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}P\left(8,4\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}P\left(3,3\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}P\left(3,3\right)=6\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}P\left(9,6\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}P\left(11,5\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}P\left(11,5\right)=55,440\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}C\left(8,5\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}C\left(12,4\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}C\left(12,4\right)=495\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}C\left(26,3\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}C\left(7,6\right)\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}C\left(7,6\right)=7\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}C\left(10,3\right)\text{\hspace{0.17em}}$

For the following exercises, find the number of subsets in each given set.

$\text{\hspace{0.17em}}\left\{1,2,3,4,5,6,7,8,9,10\right\}\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}{2}^{10}=1024\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}\left\{a,b,c,\dots ,z\right\}\text{\hspace{0.17em}}$

A set containing 5 distinct numbers, 4 distinct letters, and 3 distinct symbols

$\text{\hspace{0.17em}}{2}^{12}=4096\text{\hspace{0.17em}}$

The set of even numbers from 2 to 28

The set of two-digit numbers between 1 and 100 containing the digit 0

$\text{\hspace{0.17em}}{2}^{9}=512\text{\hspace{0.17em}}$

For the following exercises, find the distinct number of arrangements.

#### Questions & Answers

if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2
wish i knew calculus to understand what's going on 🙂
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
secA+tanA=2√5,sinA=?
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT