Permutations – Special CasesM408 Probability Unit

❑𝒏𝑷𝒏=𝒏!• Example 1 – a.) How many unique ways are there to arrange the letters PIG?

b.) How many unique ways are there to arrange the letters BOO?

𝑷𝒆𝒓𝒎𝒖𝒕𝒂𝒕𝒊𝒐𝒏𝒘𝒊𝒕𝒉𝑹𝒆𝒑𝒆𝒕𝒊𝒕𝒊𝒐𝒏• To arrange ‘n’ items with‘p’ repeats of one type, possibly ‘q’ repeats of another type, possibly others,…use

Example 2 – How many ways can you arrange the letters of…a.) ATTRACTIVE

b.) MISSISSIPPI

Example 3 –

How many ways can you arrange 5 red flags and 8 white flags in a line?

Example 4 - How many unique ways can you arrange 5 people {A,B,C,D,E} in a circle (as opposed to a line)?

A

B

CD

E

How would you name this arrangement?ABCDEBCDEACDEABDEABCEABCDThese 5 ‘arrangements’ are all the same when you arrange the items in a circle!

Circular Permutations

For ‘n’ objects in a circular arrangement, there are or (n-1)! Permutations.

Example 5 -

How many ways are there to arrange 9 people at a rectangular table?

Reference Points – Not all circular arrangements have circular permutations.If one position in the arrangement is ‘special’, we treat it as a starting point.If there is a clear reference (starting) point, then the arrangement is considered linear.

Example 6 -

How many ways are there to arrange 9 people at a rectangular table that has one really fancy chair?

Reflection PrincipleApplies to arrangements that can be turned over and viewed from a different perspective (bracelets, beaded necklaces, etc.)If an arrangement is reflective, divide your answer by 2.

Example 7 - How many ways can you arrange 6 beads on a bracelet?

AB

CD

E

FA

F

ED

C

BFlip!

These two arrangements are the same in a flipped object.

Example 8 - How many ways can you arrange 6 beads on a bracelet that has a clasp?