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<p>Find the number of $4$ -letter code-words that can be formed from the thirteen letters of the word <strong>MEDITERRANEAN</strong>.</p>

Find the number of $4$ -letter code-words that can be formed from the thirteen letters of the word **MEDITERRANEAN**.

Permutation and Combination

<p>The following diagram shows $9$ distinct point chosen from the sides of a triangle.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1758615559935-tmva74dm8l.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTg2MTU1NTk5MzUtdG12YTc0ZG04bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzIzOTI3LCJleHAiOjE3ODU3MjQyMjd9.p9DoBxpe5z2SQnwHMH3cTGZaUlSPja08ptpv4-7y-QM" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">How many line segments are there joining any two points on different sides?</div></li><li><div class="list-item-paragraph">How many triangles can be formed form these points?</div></li></ol>

The following diagram shows $9$ distinct point chosen from the sides of a triangle.

How many line segments are there joining any two points on different sides?

How many triangles can be formed form these points?

Permutation and Combination - General

Choosing points to form triangles

<p>Ava and Barry play a game containing one green marble and two red marbles. Each player intron randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that Ava wins on her first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Barry wins on his first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Ava wins in one of her first three turns.</div></li><li><div class="list-item-paragraph">Find the probability that Ava eventually wins.</div></li></ol>

Ava and Barry play a game containing one green marble and two red marbles. Each player intron randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.

Worked Example 5

<p>$3$ boys and $5$ girls and $1$ married couple are to be seated in a row. In how many ways can they be seated if</p>
<ol class="i-list"><li><div class="list-item-paragraph">the married couple sits together,</div></li><li><div class="list-item-paragraph">the boys sit next to one another,</div></li><li><div class="list-item-paragraph">the girls sit next to one another,</div></li><li><div class="list-item-paragraph">there are $2$ people between the married couple.</div></li></ol>

$3$ boys and $5$ girls and $1$ married couple are to be seated in a row. In how many ways can they be seated if

<p>Find the number of possible committees of $5$ members that can be formed form $6$ men and $5$ women if</p>
<ol class="i-list"><li><div class="list-item-paragraph">there are no restrictions,</div></li><li><div class="list-item-paragraph">there are exactly $2$ men in the committee,</div></li><li><div class="list-item-paragraph">there are more men than women in the committee,</div></li><li><div class="list-item-paragraph">the committee has at least one man and at least one woman.</div></li></ol>

Find the number of possible committees of $5$ members that can be formed form $6$ men and $5$ women if

<p>In how many ways can $8$ people be divided into</p>
<ol class="i-list"><li><div class="list-item-paragraph">groups of $2$ people and $6$ people,<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$2$ groups of $4$ people,<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$4$ groups of $2$ people,<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">${{1}^{\mathrm{st}}}$ pair, ${{2}^{\mathrm{nd}}}$ pair, ${{3}^{\mathrm{rd}}}$ pair and the ${{4}^{\mathrm{th}}}$ pair.<span class="inline-marks">[1]</span></div></li></ol>

In how many ways can $8$ people be divided into

<p>4 boys and 3 girls and 1 married couple are to be seated in a row. In how many ways can they be seated if</p>
<ol class="i-list"><li><div class="list-item-paragraph">the married couple sits together,</div></li><li><div class="list-item-paragraph">the boys sit next to one another,</div></li><li><div class="list-item-paragraph">the girls sit next to one another,</div></li><li><div class="list-item-paragraph">there are 2 people between the married couple.</div></li></ol>

4 boys and 3 girls and 1 married couple are to be seated in a row. In how many ways can they be seated if

Worked Example 4

<p>Find the number of ways to divide 16 people into</p>
<ol class="i-list"><li><div class="list-item-paragraph">two groups consisting of 7 and 9 people, </div></li><li><div class="list-item-paragraph">two groups consisting of 8 people each, </div></li><li><div class="list-item-paragraph">four groups consisting of 4 people each, </div></li><li><div class="list-item-paragraph">three groups consisting of 6, 6 and 3 people with 1 particular person excluded. </div></li></ol>

Find the number of ways to divide 16 people into

Worked Example 6

<p>Find the number of permutations using all the letters in “PEPPERONI”.</p>

Find the number of permutations using all the letters in “PEPPERONI”.

Worked Example 1

<p>A coach has, in his soccer team, $5$ strikers, $8$ midfielders, $9$ defenders and $3$ goalkeepers. He has to select exactly $1$ goalkeeper and $10$ other players to start a match. </p>
<ol class="i-list"><li><div class="list-item-paragraph">in how many ways can he select the $11$ players, </div></li><li><div class="list-item-paragraph">in each of the four sections (goalkeepers, strikers, defenders and midfielders), there is one STAR player. How many ways can the coach choose his $11$ players if he must include at least one STAR player? </div></li></ol>

A coach has, in his soccer team, $5$ strikers, $8$ midfielders, $9$ defenders and $3$ goalkeepers. He has to select exactly $1$ goalkeeper and $10$ other players to start a match. 

<p>There are five elements in the set $\{v,w,x,y,z\}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">How many possible subsets can be formed from this set?</div></li><li><div class="list-item-paragraph">How many proper subsets can be formed from this set?</div></li></ol>

There are five elements in the set $\{v,w,x,y,z\}$.

Permutations of Choices Q1

<p>Find the number of ways to choose 4 letters from the word RESETTLEMENT.</p>

Find the number of ways to choose 4 letters from the word RESETTLEMENT.

Worked Example 2

<p>At a table, 5 boys and 4 girls are to be seated on a round table. How many ways can this be done if</p>
<ol class="i-list"><li><div class="list-item-paragraph">there is no restrictions,</div></li><li><div class="list-item-paragraph">there is no restrictions and the seats are numbered,</div></li><li><div class="list-item-paragraph">4 girls must sit together,</div></li><li><div class="list-item-paragraph">3 particular boys cannot sit together?</div></li></ol>

At a table, 5 boys and 4 girls are to be seated on a round table. How many ways can this be done if

Worked Example 7

<p>John and 5 other boys are to form a queue with 3 girls. Find the number of possible arrangements if</p>
<ol class="i-list"><li><div class="list-item-paragraph">John is the first in the queue,</div></li><li><div class="list-item-paragraph">the first and the last are of the same sex.</div></li></ol>

John and 5 other boys are to form a queue with 3 girls. Find the number of possible arrangements if

Permutations with Restrictions Q1

<p>In how many ways can $5$ different books be distributed among $10$ people if each person can get any number of books?</p>

In how many ways can $5$ different books be distributed among $10$ people if each person can get any number of books?

Permutations of Choices Q9

<p>*The prime factors of $543 312$ are $2$, $3$, $7$ and $11$.  Excluding $1$ and $543 312$, how many positive factors of $543 312$ are there?<br></p>

*The prime factors of $543 312$ are $2$, $3$, $7$ and $11$.  Excluding $1$ and $543 312$, how many positive factors of $543 312$ are there?


Permutations with Objects Combined Q1

<p>Given that all balls are identical in shape and sizes. Find the number of arrangements of 3 red balls, 4 blue balls and 5 purple balls in which</p>
<ol class="i-list"><li><div class="list-item-paragraph">the red balls are separated,</div></li><li><div class="list-item-paragraph">the blue balls are separated,</div></li><li><div class="list-item-paragraph">the purple balls are separated.</div></li></ol>

Given that all balls are identical in shape and sizes. Find the number of arrangements of 3 red balls, 4 blue balls and 5 purple balls in which

<p>Find the number of arrangements of all eleven letters in the word PERSISTENCE in which</p>
<ol class="i-list"><li><div class="list-item-paragraph">R and N are separated,</div></li><li><div class="list-item-paragraph">the vowels are all separated.</div></li></ol>

Find the number of arrangements of all eleven letters in the word PERSISTENCE in which

Permutations with Objects Separated Q2

<p>How many numbers of five digits can be formed from the digits $1$, $2$, $2$, $2$, $3$, $4$, $5$?	<span class="inline-marks">[4]</span></p>

How many numbers of five digits can be formed from the digits $1$, $2$, $2$, $2$, $3$, $4$, $5$?	

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