1. Standards: 2.7.8A Determine the number of permutations for an event.
Objectives: Solve problems involving linear permutations of distinct or indistinguishable objects. Solve problems involving circular permutations.
Standards: A Determine the number of permutations for an event.

2. A permutation is an arrangement of objects in a specific order.
When objects are arranged in row,
the permutation is called a linear permutation.
You can use factorial notation
to abbreviate this product:
4! = 4 x 3 x 2 x 1 = 24.
If n is a positive integer, then n factorial,
written n!, is defined as follows:
n! = n x (n-1) x (n-2) x x 2 x 1.
Note that the value of 0! = 1.

3. I. Permutations of n Objects - the number of permutations of n objects is given by n! {factorial button – go to Math to PRB to # 4}
Ex 1. In 12-tone music, each of the 12 notes in an octave must be used exactly once before any are repeated. A set of 12 tones is called a tone row. How many different tone rows are possible?
Ex 2. How many different ways can the letters in the word objects be arranged?
12! = 479,001, 600
7! = 5040

4. Permutations of n Objects Taken r at a Time – the number of permutations of n objects taken r at a time, denoted by P(n, r), is given by P(n, r) = nPr =__n!_, where r < n (n–r)!
Ex 1. Find the number of ways to listen to 5 different CDs from a selection of 15 CDs.
Ex 2. Find the number of ways to listen to 4 CDs from a selection of 8 CDs.
Ex 3. Find the number of ways to listen to 3 different CDs from a selection of 5 CDs.
15 P 5 = 360, 360
8 P 4 = 1680
5 P 3 = 60

5. Permutations with Identical Objects – the number of distinct permutations of n objects with r identical objects is given by n!/r! where 1 < r < n. The number of distinct permutations of n objects with r1 identical objects, r2 identical objects of another kind, r3 identical objects of another kind, , and rk identical objects of another kind is given by _______n! _ r1 ! * r2 ! * r3 ! rk !

6. 11!__ (5! * 6!) Ex 1. Anna is planting 11 colored flowers in a line.
In how many ways can she plant 4 red flowers,
5 yellow flowers, and 2 purple flowers?
Ex 2. In how many ways can Anna plant 11 colored
flowers if 5 are white and the remaining ones are red?
11!__
(5! * 6!)
= 462

7. Ex 3. Frank is organizing sports equipment for the physical education room.
He has 15 balls that he must place in a line.
In how many ways can he line up 6 footballs, 2 soccer balls,
4 kickballs, and 3 basketballs?
Ex. 4 BETWEEN
____15!______
(6! * 2! * 4! * 3!)
= 6,306,300 7!
3! = 840

8. III. Circular Permutations - If n distinct objects are arranged around a circle, then there are (n – 1)! Circular permutations of the n objects.

9. Ex 2. In how many ways can seats be chosen for 12 couples
on a Ferris wheel that has 12 double seats?
Ex 3. In how many different ways can 17 students
attending a seminar be arranged in a circular seating pattern?
(12 – 1)! = 11! = 39, 916, 800
(17 – 1)! = 16! = 2.09 X 1013

11. Writing Activities

12. REVIEW OF PERMUTATIONS