**Probability Permutations And Combinations Worksheet With Answers Pdf** – 1 35 Probability, Probability, and Probability So far we have been able to calculate the elements of the product space by drawing a tree diagram. For large product areas, the construction of wood carvings has become very difficult. In this section, we discuss mathematical methods to find the number of elements in a sample space or assembly without a sequence. Substitution Consider this problem: In how many ways can 8 horses finish the race (if there is no tie)? We can consider this problem as an 8-step solution. The first step is the first in the horse race, the second step is the second for the horse,…, the 8th step is the possibility of getting the position of 8 in horse racing. Therefore, according to the Basic Principle of Counting = 40, there are 320 ways. This problem shows an example of ordering, meaning that the order of the elements in the order is important. Such an ordered arrangement is called a permutation. samples can be written in a summary called factorial. so, = 8! (Read 8 items). In general, we define n factorial by {n (n 1) (n 2) 3 2 1 if n 1 n! = 1 if n = 0. where n is a number. Example 35.1 Consider the following expressions: (a) 6! (b) 10! 7! (a) 6! = 720 (b) 10! = 720 7! The number of permutations of n factors using factorials is n! we will see. 1

2 Example 35.2 There are 6! replace the 6 letters in the word square. How many of them are the second letter? Let r be the second letter. Then there are 5 ways to the first level, 4 ways to the third place, 3 ways to the fourth place, etc. There are 5! such a replacement. Example 35.3 There are five different books on a shelf. How many different ways can they be designed? Five books can be collected with = 5! = 120 ways. Permutational Calculation We will consider the permutation of a set of elements from a larger set. Suppose we have n objects. How many ordered sequences of r elements can we make from these n elements? P (n, r) denotes the number of cases. n represents the number of different elements and r represents the number of them found in each order. This is equivalent to finding the number of permutations of people we can get with r seats if we have people to choose from. We continue like this. The first place can be filled with n people; who is among the other (n 1) second persons and so on. The fourth place can be filled with (n r + 1) people. So we can see that P (n, r) = n (n 1) (n 2)…(n r + 1) = n! (no r)!. Example 35.4 In how many ways can the gold, silver, and bronze medals be awarded for an 8-man race? Using the multiplication formula, we get P (8, 3) = 8! (83)! = 336 ways. Example 35.5 How many five-digit zip codes can there be in which all digits are unique? Possible numbers are 0 to 9. 2

## Probability Permutations And Combinations Worksheet With Answers Pdf

3P (10, 5) = 10! (105)! Working problems = 30, 240 zip codes. Problem 35.1 Evaluate each of the following expressions. (a) (2) (3) (4!) (b) (43)! (c) 43! (d) 4! 3! (e) 8! 5! (f) 8! 0! Problem 35.2 Calculate each of the following. (a) P (7, 2) (b) P (8, 8) (c) P (25, 2) Problem 35.3 Find m and n such that P (m, n) = 9! 6! Problem 35.4 If repetition is allowed, how many four-digit words can be formed using the same 26 letters (a)? b) if repetition is not allowed? Problem 35.5 Some cars have three numbers and three letters. a) How many plates can be obtained if no repetition of letters is allowed? b) How many plates are there if letters and numbers are not repeated? 3

### Permutations And Combinations Worksheet

4 Problem 35.6 A combination lock has 40 numbers. a) How many different combinations of three numbers are there? (b) How many combinations if the numbers are different? Problem 35.7 (a) Miss Murphy wants to have her 12 students line up for a class picture. How many different seats are there? (b) Seven of Miss Murphy’s students are female and five are male. In how many different ways can he place 7 women on his left and 5 men on his right? Problem 35.8 Using the digits 1, 3, 5, 7, and 9, how many digits (a) can be formed without repeating digits? (b) Can there be two-digit numbers? (c) Are three-digit numbers possible? (d) What four-digit numbers are possible? Problem 35.9 A math group has five members. How are the officers, president and treasurer elected? Problem (a) There are nine players on a basketball team. Find out how a manager can shape the batting order. (b) Find ways to select the top three letters from an alphabet if none of the letters are repeated. Integration As mentioned above, exchange deals with the arrangement of a set of things or people. However, there are many problems where we want to know the number of ways to select r items from n different items in each order. For example, when selecting a two-person committee from a 10-person group, the order of the committee does not matter. This is the election of Mr. Before Mrs. B to a 4th committee

5 Choose Ms B and Mr A. A combination is a group of things that do not differ in order. Combination Count Let C (n, r) denote the number of ways to select r items from a set of n unique items. Since the number of groups of r elements out of n elements is C(n, r) and each group can be sorted by r! then P (n, r) = r!c (n, r) means. It follows that C (n, r) = P (n, r) r! = n! r! (of r)! Example 35.6 How many ways are there to choose two slices of pizza from a plate with one slice of pepperoni, sausage, mushroom, and cheese pizza? Order doesn’t matter when choosing a pizza slice. This process is a combination. So we need to find C(4, 2) = 4! = 6. So six 2!(4 2)! Ways to choose two slices of pizza from the plate. Example 35.7 How many ways are there to select a committee that will design mathematics courses in a school if the committee consists of 3 teachers from the math department, 4 teachers from the computer science department, 9 teachers from the math department, and 11 from the CS department? C(9, 3) has C(11, 4) = 9! 3 (93)! 11! 4 (114)! Work problems = 27, 720 ways. Problem Complete each of the following: (a) C (7, 2) (b) C (8, 8) (c) C (25, 2) Problem Find m and n such that C (m, n) = 13 5 .

6 Problem The science library group recommends three books from the list 42. If you circle the three choices of the number 42 on a postcard, how many choices are there? Problem At the beginning of the second quarter of a math class for elementary teachers, each of the 25 students in the class shakes hands with each of the students exactly once. How many handshakes? Problem A math team has five members. In how many ways can a two-person social committee be chosen? Problem A consumer group plans to select 2 out of 8 televisions to test picture quality. In how many ways can they choose 2 TV sets? Problem A chess club has six members. In how many ways (a) can all six terms be arranged in this picture? (b) can they elect a president and secretary? (c) Can three members choose to participate in regional competition regardless of order? Problem C(m, n) = P (15, 2) Problem A school has 30 teachers Find the minimum value of m and n. In how many ways can a school choose 3 people to participate in the National Assembly? Problem Which is greater, the number of combinations or the number of permutations of a set of elements? The problem is how many different judges of 12 people can be chosen from a group of 20? 6

7 Looking for possibilities

### Probability With Permutations And Combinations

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