# 9.5 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.

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.

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar