Combinations Example:

Six men and seven women apply for two identical jobs. If the jobs are filled at random, find the probability both positions are filled by men.

Step 1:

As the jobs are idenitical we see that we will use combinations as order is not important 

Step 2:

Calculate the size of the sample space.

Here our sample space is all the ways which we can fill these two positions out of the 10 people that applied for the that is

\[_{13}C_2=78\]

Step 3: 

Calculate the size of the event

Here our event is all the ways which we can fill these two positions with only men

\[_6C_12=15\]

Step 4:

Calculate the probability

\[P(\text{filling both positions with men})=\frac{_6C_2}{_{13}C_2}=\frac{15}{78}\approx 0.19\]

---

Permutations Example:

If the letters of Indianapolis are chosen in a random order what is the probability it is spelled correctly?

Step 1:

As the orders of the letters is important when spelling a word correctly we see that order matters and thus we should use permutations.

Step 2:

Calculate the size of the sample space.

Here our sample space is all the ways which we can fill lay out in order all the letters of the word Indianapolis (12 letters)

\[_{12}P_{12}=12!=479001600\]

Step 3: 

Calculate the size of the event

Here our event is all the ways which we could pull indianapolis in order, to count this we see that we actually have \(_3P_3\) choices how we can place the 3 I's and \(_2P_2\) choices how we can place the 2 N's as well as \(_2P_2\) choices how we can place the 2 A's, that is we have the following number of correct permutations

\[(_3P_3)\cdot(_2P_2)\cdot(_2P_2)=(3!)\cdot(2!)\cdot(2!)=24\]

Step 4:

Calculate the probability

\[P(\text{filling both positions with men})=\frac{(_3P_3)\cdot(_2P_2)\cdot(_2P_2) }{_{12}P_{12} }=\frac{24}{479001600 }\]