Permutations and Combinations

Lesson

To work out the probability of something occurring we consider the two values: the number of ways the thing you want can happen and the total number of ways it could happen

Probabilities can be represented as a **fraction **-

$\text{Probability of Something Occurring}=\frac{\text{What you want}}{\text{Total possible}}$Probability of Something Occurring=What you wantTotal possible

Or as a **percentage **-

$\text{Probability of Something Occurring}=\frac{\text{What you want}}{\text{Total possible}}\times100%$Probability of Something Occurring=What you wantTotal possible×100%

We still approach the calculations of the combinations the same, but then we have the added step of calculating the probabilities.

A magazine editor is deciding which of $5$5 articles to print as the cover story on the front page.

*If only *$2$2* stories can be chosen, how many different selections are possible?*

First, we confirm that it is a combination, not a permutation in that the order DOESN'T matter. So we know now we need to use the combination formula. We check the question looking for restrictions, but there isn't any.

$C(n,r)=\frac{n!}{r!(n-r)!}$`C`(`n`,`r`)=`n`!`r`!(`n`−`r`)!

Where $n$`n` is the total number of articles $n=5$`n`=5 and $r$`r` is the number we are choosing, $r=2$`r`=2.

$C(5,2)=\frac{5!}{2!(5-2)!}$`C`(5,2)=5!2!(5−2)!

$C(5,2)=\frac{5!}{2!(3)!}$`C`(5,2)=5!2!(3)!

$C(5,2)=\frac{5\times4\times3!}{2!3!}$`C`(5,2)=5×4×3!2!3!

$C(5,2)=\frac{5\times4}{2!}$`C`(5,2)=5×42!

$C(5,2)=10$`C`(5,2)=10

So there are $10$10 combinations.

*If you were the writers of one of the stories, what is the probability that your article is chosen for the front page?*

If your article is chosen, then it is definitely one of the two. So there are now ^{4}C_{1} possible combinations of your article (and one other) being on the front cover.

Total combinations of your article appearing = $C(4,1)=4$`C`(4,1)=4

Thus the probability becomes,

$\text{Probability of Your Article}=\frac{\text{What you want}}{\text{Total possible}}\times100%$Probability of Your Article=What you wantTotal possible×100%

$\text{Probability of Your Article}=\frac{4}{10}\times100%=40%$Probability of Your Article=410×100%=40%

Or you could write it as $\frac{2}{5}$25 to leave the probability as a fraction

A box contains 6 pens of different colours: red, green, blue, yellow, black and white. Two pens are drawn at random without replacement.

How many possible selections are there?

What is the probability of drawing the green and black pens?

A menu has three entrées ($E_1,E_2,E_3$`E`1,`E`2,`E`3), four mains ($M_1,M_2,M_3,M_4$`M`1,`M`2,`M`3,`M`4) and two desserts ($D_1,D_2$`D`1,`D`2). A meal is made up of one of each.

How many different meals are possible?

What is the probability of selecting $E_1$

`E`1 , $M_3$`M`3 and $D_2$`D`2?How many different meals are possible given that $E_1$

`E`1 is the entrée?

$5$5 people are to be selected from a larger group of $10$10 candidates. If Amelia is among the candidates, what is the probability that she will be among those selected?

Use permutations and combinations