Permutations with repetitions

n objects of which p are alike

and q are alike !

! !

n

p q

PERMUTATIONS

How many different ways can letters of each word be arranged

•1. trigonometry

•2. algebra

•3. probability

Arrange objects

• 2 red lights, 4 yellow lights, 5 blue lights, 1 green light, and 2 white lights

How many ways can you arrange on a string of lights?

• How many different zip codes can have the digits

2, 8, 3, 3, 3?

Circular permutations

• If n objects are arranged in a circle, then there are or

!n

n( 1)!n

Circular permutation turned linear

• If n objects are arranged relative to a fixed point, then there are n! permutations.

• In other words, circular arrangements with fixed points of reference are treated as linear permutations. n!

Circular or LinearHow many arrangements?

1. Nine bowls of food are placed on a revolving tray in the center of the table.

2. Seven people are to be seated at a round table where one person is seated next to the window.

Circular or Linear

• 3. 8 jewels on a necklace

• 4. 5 beads strung on a string in a square pattern relative to a knot in the string

• 5. 12 gondolas on a Ferris wheel

• 6. a stack of 6 pennies, 7 dimes, and 10 quarters

Circular or Linear• 7. 6 houses of a cul-de-sac relative

to the incoming street

• 8. 5 points on a circle, how many ways can the points be named on the circle using the letters A-E?

• 9. 5 points on a circle, named A-E, relative to the point on the x-axis, how many arrangements are possible?

Worksheet 13.2 odd