$1\mathrm{.}5$ Assignment: Organized counting

In $1-3$ sentences, define each of the following terms in your own words and provide an example:

Permutation Bold text start$(2$ marks$)$Bold text End

Combination Bold text start$(2$ marks$)$Bold text End

Factorial Bold text start$(2$ marks$)$Bold text End

Mutually exclusive Bold text start$(2$ marks$)$Bold text End

Chris $($they$/$them$)$ is planning a meal for a party. The meal will include an appetizer, a main course, and dessert. They have three choices for an appetizer $($Artichoke$,$ Bruschetta, or Caviar$),$ two choices for a main course $($Drumsticks or Eggplant$),$ and three choices for dessert $($Fritter$,$ Gelato, or Hot chocolate$).$

Create a tree diagram that depicts all possible options for Chris$$ meal. You may create your diagram on the computer, or draw it on paper using a ruler and dark pen and then insert a clear photograph or scan into your assignment submission. Bold text start$(3$ marks$)$Bold text End

How many different meals are possible? Bold text start$(1$ mark$)$Bold text End

How many different meals are possible if Chris cannot serve Eggplant and Fritter together? Explain your answer. Bold text start$(2$ marks$)$Bold text End

How many different meals are possible if Chris must serve at least one of Artichoke, Eggplant, and Hot chocolate? Explain your answer. Bold text start$(3$ marks$)$Bold text End

A teacher is creating a test from a question bank. If question $1$ can be chosen from $8$ options, question $2$ chosen from $3$ options, question $3$ from $2$ options, and question $4$ from $21$ options, use the multiplicative counting principle to determine the number of different tests the teacher could make. Bold text start$(3$ marks$)$Bold text End

Given the $8$ letters in the word ABSOLUTE,

How many ways can the letters be rearranged? Bold text start$(2$ marks$)$Bold text End

How many arrangements can be formed if A and B must be together? Bold text start$(3$ marks$)$Bold text End

How many arrangements can be formed if the word must start with a vowel $($A$,$ E$,$ O$,$ or U$)$ and end with $$A$$ or $$T$?$ Bold text start$(5$ marks$)$Bold text End

Given the digits $\{1,2,3,4,5\},$ how many $3-$digit numbers can be formed if:

No digit is repeated? Bold text start$(2$ marks$)$Bold text End

The result must be even, and no digit is repeated? Bold text start$(2$ marks$)$Bold text End

The number must contain at least $1$ repeated digit? Bold text start$(3$ marks$)$Bold text End

$6.$ A deck of cards in a game contains all four suits $($Clubs $$ and Spades $,$ which are black, and Hearts $$ and Diamonds $,$ which are red$),$ but only the cards with values $5,6,7,8,9,$ and $10.$

How many different $5-$ card hands can be formed using this deck? Bold text start$(3$ marks$)$Bold text End

How many different $5-$card hands can be formed that contain at least $4$ red cards? Bold text start$(4$ marks$)$Bold text End

How many different $5-$card hands can be formed that contain at least one club and at least one heart? Bold text start$(4$ marks$)$Bold text End

How many different $5-$card hands can be formed that contain at least one spade and at least two $10$s$?$ Bold text start$(5$ marks$)$Bold text End

Write row $8$ of Pascal$$s triangle, and explain how you found it$.$ Bold text start$(3$ marks$)$Bold text End

Write the $11$th row of Pascal$$s triangle and explain how you found it$.$ Bold text start$(3$ marks$)$Bold text End

Consider the Cartesian $($x$-$y$)$ grid. If you can only move right and up$,$ how many paths are there:

From A$(0,0)$ to B$(6,3)?$ Bold text start$(4$ marks$)$Bold text End

From A$(0,0)$ to C$(9,10)$ if you must pass through D$(5,4)$ and E$(7,8)?$ Bold text start$(5$ marks$)$Bold text End

$10.$ In a survey of $200$ MDM$4$U students, the following information was found:

$68$ enjoy crosswords,

$97$ enjoy baking,

$71$ enjoy acting,

$20$ enjoy acting and crosswords,

$29$ enjoy acting and baking,

$23$ enjoy baking and crosswords,

$28$ do not enjoy any of those three activities.

Explaining any calculations,

How many people enjoy all three activities? Bold text start$(4$ marks$)$Bold text End

How many people enjoy only acting? Bold text start$(2$ marks$)$Bold text End

How many people enjoy only baking? Bold text start$(2$ marks$)$Bold text End

Create a Venn diagram that depicts the number of students who enjoy the activities. You may draw your diagram and scan or photograph it$,$ you may create it electronically, or if you are unable to complete this activity visually, you may describe the components of the diagram in detail. Bold text start$(5$ marks$)$Bold text End