2–Quizz 1–Groupwork Combinatorics

Season 01 • Episode 12 • Combinatorics

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Combinatorics

Season 01

Episode 12 Time frame 1 period

Prerequisites :

^Eventsandprobabilities

Objectives :

•

^{Disover afew} ombinatoris methods

Materials :

•

^{Exerise sheet for eahstudent.}

•

^{Listof questions for the quizz.}

1 – Group work 25 mins

Students work in groups of 4. They have to solve an exerise involving two standard

ombinatoris situationsand the fatorialnotation.

2 – Quizz 30 mins

Students arestillworking ingroups.They havetoanswertensimple ombinatorisques-

tions (3 minutes per question). At the end of the quizz, grouppapers are marked by the

Combinatorics

Season 01

Episode 12 Document Exercise

There are 26 letters in the latin alphabet we use, from a to z. A word is an ordered

sequene ofletters,withpossiblerepetitions.Intherstpart ofthis exerisewewillount

the number of possible words, without onsidering the fat that a word asa meaning or

not. For example, yyyyryyyr is a9-letters word, just like pentagone.

1

^{. Howmany possible 2-letterswords are there?}

2

^{. Howmany possible 3-letterswords are there?}

3

^{. Howmany possible 4-letterswords are there?}

4

^{. Find ageneralformulafor the number ofpossible}

n

-letters words, with

n

anatural number.

5

^{. The longest English word appearingin a ditionnary is}

Pneumonoultramirosopisiliovolanooniosis

It'sadiseaseofthelungs,ausedbyinhalingmineralormetallidust,suhassilion

and quartzite.

Count the number of possiblewords with the same number of letters.

6

^{. The totalnumberofpossiblewords with46lettersof less is}approximatelyequalto

1.275872703 × 10 ⁶⁵

. Imagine that we want to reate a ditionnary listing all these possible words, and that we manageto put 500 words in eah page. Now, suppose

we use a paperwith a thikness of 0,01 millimeters(whih is very thin).

Compute the number of pages needed and total thikness of the ditionnary,using

the appropriate unit.

In the seond part, we will study a dierent problem : the number of anagrams for a

given word. An anagram is the result of rearranging the letters of a word or phrase to

produe anew word or phrase,using allthe original letters exatly one. An interesting

example isI'm adot in plae,ananagramof A deimalpoint; afamous one is I Am

Lord Voldemort, an anagram of Tom Marvolo Riddle. In the following questions, we

willonsider allanagrams,whether they mean somethingornot. Forexample,fntnuio

is ananagram offuntion.

1

^{. List allthe anagrams of the words no, yes and blue.}

2

^{. Find outthe number ofanagrams ofany 2-lettersword, any 3-lettersword andany}

4-letters word, ounting ineah ase the word itself.

3

^{. The notation}

n!

denotes the produt of all integers from

1

to

n

. Use this notation to write the answers of the previous question.

4

^{. Prove that the number of anagrams of any 5-lettersword is}

5!

.

5

^{. What is the numberof anagrams of any}

n

-letters word.

Season 01 • Episode 12 • Combinatorics

²

Document 1

^Quizz

1

^{. For every meal, a student has to hoose between 6 starters, two main ourses, two}

sides, 5 heeses oryoghurt and 3 desserts. What isthe number of possible meals?

2

^{. A man wears only blak and white. Every morning he piks randomly one pair of}

shoes (blak or white), one pair of trousers (blak or white), one shirt (blak or

white) and one overoat (blakor white).What isthe numberof possibleoutts?

3

^{. In horseraing} terminology,atrifeta isa parimutuel betinwhihthe bettor must predit whih horses will nish rst, seond, and third in exat order. If there are

12 horsesin the rae, how many dierent hoies are there?

4

^{. Every morning, a man living in London piks randomly an umbrella before going}

out inthe rain.He anhoose between threedierentones :one green,one redand

one blak. What is the numberof possible hoies for one week.

5

^{. MarioKart Wii is a famous rainggame developed by Nintendo.In this game,the}

player must hoose between 4 modes, 24 Nintendo haraters, 8 vehiules for eah

harater, 8ups and 4 raes per up. What is the number of possible raes?

6

^{. A group of120 people have toelet a board madeof a Chairman,a} vie-Chairman and a seretary. Howmany possible boards are there? Anyone an be eleted, but

a dierent person must be eletedfor eahpost.

7

^{. Everyweek,ateaherpiksrandomlyapupilinalassof36towipeawaytheboard}

after eah lesson. What is the number of possible hoies for 5weeks?

8

^{. Ten peoplehavetobearrangeinalineandhanded outnumbers from1to10.What}

is the number of possible arrangements?

9

^{. How many distint ar liene plates are there onsisting of seven haraters, the}

rst two and lasttwo being letters and the middle ones digits?

10

^{. As I was goingto StIves}

I met a man with seven wives

Eahwife had seven saks

Eahsak had seven ats

Eahat had seven kits

Kits, ats, saks, wives