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Combinations and Permutations : Six Problems

1) Give clearly justified answers to the following. a) How many 7-digit telephone numbers can be formed if the first digit cannot be 0 or 9 and if the last digit is greater than or equal to 2 and less than or equal to 3? Repeated digits are allowed. b) How many different ways are there to arrange the 6 letters of the word CA


Define the sign of a permutation $ to be: sgn $ = 1 if $ is even. -1 if $ is odd. Prove that sgn($%) = sgn$sgn% for all $ and % in Sn.

Permutations : Parity

Show that $ and %$%^-1 have the same parity for all % and $ in Sn. (Sn is the symmetric group of degree n)


Q: How many four-digits numbers can be formed under the following conditions? (a) Leading digits cannot be zero. (b) Leading digits cannot be zero and no repetition of digits is allowed. (c) Leading digits cannot be zero and the number must be a multiple of 5.


Find the number of inversions in the following permutation: (3, 2, 1)

Order of a Permutation

Please see the attached file for the fully formatted problems. Find the order of sigma^1000, where sigma is the permutation (123456789) (378945216) Find the order of , where is the permutation . Solution. Since and . Let ,

Sets: Element of Q

Show that if a,b is an element of Q then a+b is an element of Q and ab is an element of Q.

Combinations, Permutations and Truth Tables

1. (a) How many license plates can a state produce if the plates can contain 6 characters (from 26 letters and 10 digits) if they can only use one digit? (b) How many ways can Mr. Paul choose 6 students from a class of 15 Boys and 12 Girls, if he must choose at least 5 boys? (c) How many orderings are there of the letter

Combinations : Seating Arrangements at a Dinner Table

In how many ways can 6 couples be seated at a circular table if each couple is not to be separated? How many ways can 5 Manchester United and 8 Chlesea players be seated at a circular dinner table if no two Manchester United players can sit together?

Permutations and combinations

A set of 10 flags, 5 red, 3 blue and 2 yellow are to be arranged in a line along a balcony. If flags of the same colour are INDISTINGUISHABLE, find the number of arrangements in which, 1) The three blue flags are together 2) The yellow flags are not together 3) The red flags occupy alternate positions in the line 4) If the

Number Patterns : Door Problem

A school had a very unusual tradition involving its 1000 students and its 1000 lockers. On opening day, after the head of the school had closed all the lockers, a student walked by and opened every single one. A second student then closed every second one (#2, 4, 6, 8 etc). A third student then changed every third locker (#3,

Permutation and Combination

For each new employee, a company gives a five-digit identification card. Each digit can be 0, 1, 2, or 3. If repetitions are allowed, how many different cards are possible. A. 1024 B. 625 C. 25 D. 500

Permutation -Combination

An anthropologist discovers an isolated tribe whose written alphabet contains only six letters called a, b, c, d, e, and f. The same letters cannot be used twice in the same word. If each different sequence of letters constitutes a different word in the language, what is the maximum number of six-letter words that the language

Working with combinations to select a baseball team.

The coach of the Morton Valley Softball Team has 6 good hitters and 8 poor hitters. He chooses 3 hitters at random. (a) In how many ways can he choose 2 good hitters and 1 poor hitter? (b) In how many ways can he choose 3 good hitters? (c) In how many ways can he choose at least 2 good hitters?

Working with permutations and combinations.

If the number of permutations of n objects taken r at a time is six times the number of combinations of n objects taken r at a time, determine the value of r. Is there enough information to determine the value of n? Why or why not?

Permutation, combination, probability

1. How many ways are there to choose 5 items (with repetition) from among 7 varieties? Explain, as precisely as possible, why we should not factor out by the number of arrangements of the 5 choices. ( Don't prove that the answer is wrong-we already did that- explain why it's wrong, i.e., why we shouldn't factor out by r!) It mig

Finding the maximum numbers of ways to arrange a word.

An anthropologist discovers an isolated tribe who has written an alphabet that contains only six letters (call the letters A, B, C, D, E, and F). The tribe has a taboo against using the same letter twice in the same word, it's never done. If each different sequence of letters constitutes a different word in the language, what is

Working with combinations.

An admission test given by a university contains 10 true-false questions. Eight or more of the questions must be answered correctly in order to be admitted. (a) How many different ways can the answer sheet be filled out? (b) How many different ways can the answer sheet be filled out so that 8 or more questions are answer

Working with sequential ambiguity.

In a specific sequence to create, a1 has the elements x1 and y1, a2 has the elements x2 and y2, a3 has the elements x3 and y3, The relationship between each term cannot be bijective, however it has a bounded range of finite whole numbers. Create a rule such that f(x1,y2)=x2 and f(x2,y3) = x3. The rule must be su

Setting up constraints.

How do I set up the contraints? A firm is engaged in the production of three types of products A, B, and C. The products earn profit of $3, $5 and $4 per unit respectively. Each product goes through three operations: cutting, sewing, and inspection. During the next period, the firm has 300 hours of cutting, 300 hours of se