Unit 6

Permutations, Combinations, and The Binomial Theorem

Permutations with Repeating Objects

In how many ways can the letters in the word “SPELL” be arranged?

Does it matter that there are two letter Ls?

If one letter L is assigned the colour green and the other letter L is assigned the colour red, then yes, the order does matter and the two letter Ls are distinguishable.


is different than


If the two Ls are considered identical, the repetition of the letter L must be considered when determining the number of permutations or arrangements of the letters in the word.

There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» letters in total.

Of those five letters, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» are the letter L.

There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mrow»«/mstyle»«/math» ways to arrange five letters. There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«mi mathvariant=¨normal¨»!«/mi»«/mstyle»«/math» ways to arrange the repeated letter L.

The number of ways to arrange the letters in the word SPELL is


There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»60«/mn»«/mstyle»«/math» possible arrangements of the letters in the word SPELL.

Pathways

The number of ways to move through a grid can be determined by thinking of direction and possible choices.

If you must move from point A to point B by moving only down or to the right, how many ways could you go?


To move from A to B you will need to move right «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» units and down «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» units. This is a total of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»8«/mn»«/mstyle»«/math» units. Notice that any order of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» moves right and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» moves down will take you to B. In other words, every move down can be considered identical, and every move right can be considered identical.



There are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»56«/mn»«/mstyle»«/math» ways to move from point A to B by only moving down or right.

Permutations with Constraints

Sometimes constraints need to be accounted for when determining a number of permutations.