# 13.5 Counting principles  (Page 7/12)

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The letters in the word “juggernaut”

The letters in the word “academia”

$\text{\hspace{0.17em}}\frac{8!}{3!}=6720\text{\hspace{0.17em}}$

The letters in the word “academia” that begin and end in “a”

The symbols in the string #,#,#,@,@,$,$,$,%,%,%,% $\text{\hspace{0.17em}}\frac{12!}{3!2!3!4!}\text{\hspace{0.17em}}$ The symbols in the string #,#,#,@,@,$,$,$,%,%,%,% that begin and end with “%”

## Extensions

The set, $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ consists of $\text{\hspace{0.17em}}\text{900,000,000}\text{\hspace{0.17em}}$ whole numbers, each being the same number of digits long. How many digits long is a number from $\text{\hspace{0.17em}}S?\text{\hspace{0.17em}}$ ( Hint: use the fact that a whole number cannot start with the digit 0.)

9

The number of 5-element subsets from a set containing $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ elements is equal to the number of 6-element subsets from the same set. What is the value of $n?\text{\hspace{0.17em}}$ ( Hint: the order in which the elements for the subsets are chosen is not important.)

Can $C\left(n,r\right)$ ever equal $P\left(n,r\right)?$ Explain.

Yes, for the trivial cases $r=0$ and $r=1.$ If $r=0,$ then $C\left(n,r\right)=P\left(n,r\right)=1\text{.\hspace{0.17em}}$ If $r=1,$ then $r=1,$ $C\left(n,r\right)=P\left(n,r\right)=n.$

Suppose a set $A$ has 2,048 subsets. How many distinct objects are contained in $A?$

How many arrangements can be made from the letters of the word “mountains” if all the vowels must form a string?

$\text{\hspace{0.17em}}\frac{6!}{2!}×4!=8640\text{\hspace{0.17em}}$

## Real-world applications

A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back.

1. How many arrangements are possible with no restrictions?
2. How many arrangements are possible if the parents must sit in the front?
3. How many arrangements are possible if the parents must be next to each other?

A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?

$6-3+8-3=8$

In horse racing, a “trifecta” occurs when a bettor wins by selecting the first three finishers in the exact order (1st place, 2nd place, and 3rd place). How many different trifectas are possible if there are 14 horses in a race?

A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?

$\text{\hspace{0.17em}}4×2×5=40\text{\hspace{0.17em}}$

Hector wants to place billboard advertisements throughout the county for his new business. How many ways can Hector choose 15 neighborhoods to advertise in if there are 30 neighborhoods in the county?

An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?

$\text{\hspace{0.17em}}4×12×3=144\text{\hspace{0.17em}}$

How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of $\text{\hspace{0.17em}}8\text{\hspace{0.17em}}$ freshmen and $\text{\hspace{0.17em}}11\text{\hspace{0.17em}}$ juniors?

How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?

$\text{\hspace{0.17em}}P\left(15,9\right)=1,816,214,400\text{\hspace{0.17em}}$

A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestra’s 10 cellists and 16 violinists in order of musical proficiency. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?

A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers—different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?

$C\left(10,3\right)×C\left(6,5\right)×C\left(5,2\right)=7,200$

A skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?

Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?

$\text{\hspace{0.17em}}{2}^{11}=2048\text{\hspace{0.17em}}$

A car wash offers the following optional services to the basic wash: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. How many washes are possible if any number of options can be added to the basic wash?

Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?

$\text{\hspace{0.17em}}\frac{20!}{6!6!8!}=116,396,280\text{\hspace{0.17em}}$

How many unique ways can a string of Christmas lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs?

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
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