Probability Factorial Permutations Combinations Week 6 TEST 2

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

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.

Factorial Notation • 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! =

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

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?

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?

What about: MISSISSIPPI?

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)

P n r Formula: n Pr = n!/(n-r)! (ex 1) 6 P 4 = 6!/(6 -4)! = 6!/2! = (6*5*4*3*2!)/2! = = 360 (ex 2) 8 P 3 = (ex 3) 5 P 5 = (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 P 3 = 5· 4· 3 = 60

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 P 3 = 1, 716 ways

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. • 1 st = $5, 000; 2 nd = $3, 000; 3 rd = $1, 000 • • With 26 students in the class, find the following: a) P(1 st Place) = b) P(Winning) = c) How many different arrangements of winners can be made from our class?

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?

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

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.

C n r Formula: n Cr = n!/(r!(n-r)!) (ex 1) 6 C 4 = 15 (ex 2) 9 C 3 = (ex 3) 5 C 5 = (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 C 3 = 10

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.

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?

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 C 3 =

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?

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?