SECTION 4 6 COUNTING Counting How many ways
- Slides: 34
SECTION 4. 6 COUNTING
Counting How many ways a situation can occur.
EXAMPLE You want to buy a Nissan Altima, so you go to their website and these are your choices 2. 5 -liter 4 -cylinder engine or 3. 5 -liter V 6 engine Regular, sports or luxury package Exterior Color : Silver, pearl white, gun metallic, java metallic, red blue or black Interior Color : Beige or Charcoal How many different possibilities are there?
EXAMPLE
COUNTING RULE If one event has m possible outcomes and a second independent event has n possible outcomes, then there are m x n total possible outcomes for the two events together.
EXAMPLE How many different license plates can be generated if the first 4 characters have to be letters and the last 3 characters have to be numbers?
EXAMPLE How many different ways can 6 different books be arranged on a shelf?
FACTORIAL SYMBOL The factorial symbol (!) denotes the product of decreasing positive whole numbers. for n > 0 n! = n * (n-1) * (n-2) * (n-3) * … * (1) for n = 0 0! = 1
FACTORIAL EXAMPLE 5! = 7! = 10! =
FACTORIAL EXAMPLE 7! / 5! = 2! * 6! 4! * 4! 23! / 18!
FACTORIAL RULE The number of permutations (order counts) of n different items when all n of them are selected.
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if there are no restrictions.
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if there are no restrictions. n = 1+1+2+3 = 7 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 7! = 5040
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if all the females stand together.
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if all the females stand together. Grouping women we will create a m and a n m = grouping women together = 4 n = order of the photograph = 4 mn = 4! 4! = 576
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if the parents stand together.
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if the parents stand together. Grouping parents we will create a m and a n m = parents together = 2 n = order of the photograph = 6 mn = 2! 6! = 1440
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if the parents do not stand together.
EXAMPLE A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if the parents do not stand together. don't stand together = total – stand together don't stand together = 5040 – 1440 don't stand together = 3600
EXAMPLE The Teknomite Corporation must appoint a president, chief executive officer (CEO), and a chief operating officer (COO). There are eight qualified candidates. How many different ways can the officers be appointed?
EXAMPLE The Teknomite Corporation must appoint a president, chief executive officer (CEO), and a chief operating officer (COO). There are eight qualified candidates. How many different ways can the officers be appointed? __ __ __ 8*7*6 336
WHAT IS THE DIFFERENCE What is the difference between these two examples? The Teknomite Corporation must appoint a president, chief executive officer (CEO), and a chief operating officer (COO). The Teknomite Corporation must appoint three planning committee members.
WHAT IS THE DIFFERENCE The Teknomite Corporation must appoint a president, chief executive officer (CEO), and a chief operating officer (COO). Order does matter. ABC ≠ BAC ≠ ACB ≠ BCA ≠ CAB ≠ CBA Positions mean something. The Teknomite Corporation must appoint three planning committee members. Order does not matters ABC = BAC = ACB = BCA = CAB = CBA No matter how they are chosen they become
EXAMPLE The Teknomite Corporation must appoint three planning committee members. There are eight qualified candidates. How many different ways can the members be appointed?
EXAMPLE The Teknomite Corporation must appoint three planning committee members. There are eight qualified candidates. How many different ways can the members be appointed? Need to multiply and divide the repeats. (8 * 7 * 6) / 6 56
KEY WORDS We can label the last two types of examples. PERMUTATION COMBINATION
PERMUTATION RULE The number of different permutations (order counts) when n different items are available, but only r of them are selected without replacement.
COMBINATION RULE The number of different combinations (order does not count) when n different items are available, but only r of them are selected without replacement.
PERMUTATION vs COMBINATION Questions The fruit salad consist of apples, grapes and bananas. Does the order we place the fruits in matter? The combination to the safe is 472. Do we care the order? Will 724 work? Will 247?
COMBINATION vs PERMUTATION Questions The fruit salad consist of apples, grapes and bananas. Does the order we place the fruits in matter? COMBINATION The combination to the safe is 472. Do we care the order? Will 724 work? Will 247? PERMUTATION
EXAMPLE A fan of Lady Antebellum music plans to make a custom CD with 12 of there 27 songs. How many different combinations of 12 songs are possible?
EXAMPLE A fan of Lady Antebellum music plans to make a custom CD with 12 of there 27 songs. How many different combinations of 12 songs are possible? Combination 17, 383, 860 combinations
EXAMPLE When four golfers are about to begin a game, they often toss a tee to randomly select the order in which they tee off. How many choices are there? What is the probability that they are arranged in alphabetical order?
EXAMPLE When four golfers are about to begin a game, they often toss a tee to randomly select the order in which they tee off. How many choices are there? Permutation = 24 What is the probability that they are arranged in alphabetical order? 1 / 24 =. 0416666666
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