1. Factorial, Permutations, Combinations Week 6 TEST # 2 – Next week!
Probability
Factorial, Permutations, Combinations
Week 6
TEST # 2 – Next week!

2. Permutations
are arrangements of n (number of objects) in a specific order.
With permutations “order matters”!
Problems involve the words: order, different, arrange, specific, place, position, rank, anything that has to do with a specific spot.

3. Factorial Notation Formula: n! = n*(n-1)*(n-2)*…..*(1)
Is a shorthand way to express multiplication of decreasing, consecutive integers.
Formula: n! = n*(n-1)*(n-2)*…..*(1)
(Ex. 1) 5! = 5*4*3*2*1 = 120.
(Ex. 2) 0! = 1
(Ex. 3) 9! =

4. Uses for Factorial:
How many different ways can I arrange the letters in my first name: JOE ?
JOE, JEO, OJE, OEJ, EJO, EOJ = 6ways
There are three letters, thus, 3!
3! = 3*2*1 = 6 ways.

5. More Examples:
(Ex. 1) How many ways can a coach arrange a line-up of 6 baseball players?
(Ex. 2) In how many different ways can I re-arrange the seating of 8 people?
(Ex. 3) How many different ways can I arrange 10 questions on a quiz?

6. More Examples:
(Ex. 4) How many different ways can I arrange the letters in the word MATH?
(Ex. 5) How many different ways can I arrange the letters in the word PASS?
(Ex. 6) How many different ways can I arrange the letters in the word TEXTBOOK?
(Ex. 7) How many different ways can I arrange the letters in the word STATISTICS?

7. What about:
MISSISSIPPI?

8. Smaller Arrangements of Larger Group
Permutation Rule – is the arrangement of n objects in a specific order, using only r at a time.
The notation for a permutation:  n Pr    n  is the total number of objects    r   is the number of objects chosen (want)

9. n Pr Formula: n Pr = n!/(n-r)!
(ex 1) 6 P4 = 6!/(6-4)! = 6!/2! = (6*5*4*3*2!)/2! =
= 360
(ex 2) 8 P3 =
(ex 3) 5 P5 =
(ex 4) How many different 3-digit numerals can be made from the digits  4, 5, 6, 7, 8   if a digit can appear just once in a numeral?         5 P3  =   5·4·3  =  60          

10. Sabres Line-up:
In how many ways can Lindy Ruff arrange 13 forwards in front lines of 3 players?
(Order matters here because there is a center, a right wing, and a left wing.)
13 P3 = 1,716 ways

11. Statistics Prize Money!
It’s time for the big pay-out! The college is going to pay-out three places to students in Thursday night Statistics.
1st = $5,000; 2nd = $3,000; 3rd = $1,000
With 26 students in the class, find the following:
a) P(1st Place) =
b) P(Winning) =
c) How many different arrangements of winners can be made from our class?

12. Special Arrangements:
How many different license plates can NY State issue if they are to have 3 letters followed by 4 numbers?
How many different license plates can NY State issue if they are to have 3 different letters followed by 4 different numbers?

13. Special Arrangements:
A new area code is being created. How many phone numbers are being created if the following specifications are met?
The 1st number cannot be a ZERO or a ONE
The first three cannot be 911 or 411.

14. Combinations
Combination:  A set of objects in which position (or order) is NOT important.
How many different groups of 3 can be formed including Deb, Lydia, and Jessica?
(3 people)
In a combination, the trio of Deb, Lydia, and Jessica is THE SAME as Jessica, Lydia, and Deb. Thus, there is only one group.

15. n Cr Formula: n Cr = n!/(r!(n-r)!) (ex 1) 6 C4 = 15 (ex 2) 9 C3 =
(ex 4) How many different 3-letter combo’s can be made from the letters  A, B, C, D, E   if a letter can appear just once in a combo?         5 C3   =  10          

16. Permutation versus Combination
What’s the diff’?
Permutation       versus       Combination
1. Picking a team captain, pitcher, and shortstop from a group.
1. Picking three team members from a group.
2.  Picking your favorite two colors, in order, from a color brochure.
2.  Picking two colors from a color brochure.
3.  Picking first, second and third place winners.
3.  Picking three winners.

17. Prize Winners
(Ex 1) – A raffle has 20 entries. The prizes include 5 gift certificates, all for $20 each. How many different groups can be selected to claim the prizes?
(Ex 2) – 15 people placed their names in a hat to win trip to beautiful downtown Sanborn. If the prize commission is only choosing 8 winners, how many groups can be formed?

18. Sabres Line-up:
In how many groups can Lindy Ruff arrange 13 forwards in front lines of 3 players?
(If order does not matter, each player could change up to be a center, a right wing, and a left wing.)
13 C3 =

19. Form a committee…
(Ex 1) – A committee is to be formed consisting of 3 people. There are 5 people to choose from, how many different committees can be created?
(Ex 2) – A committee is to be formed consisting of 5 people. There are 12 people to choose from, how many different committees can be created?

20. Special Emergency Committee
A new committee is to be formed from a group of 20 college students. It is to have 6 members and it must contain an equal amount of boys as girls. There are 12 boys in the original group. How many groups can be formed?