# 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?

June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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