MATH 30-1



MATH 30-1Permutations, Combinations & Binomial Theorem Module EightModule / Unit 8 - Assignment Booklet Student: __________________________________________________ Date Submitted: ___________________________________________ Lesson 1: Fundamental Counting Principle1.In Alberta, all postal codes start with the letter T. How many different postal codes are possible in Alberta?In Canada, all postal codes are made up of six characters, which follow the pattern letter-digit-letter-digit-letter-digit. The first letter of the code identifies a specific region. Each of the remaining two letters in the postal code can be any of 20 letters, and each of the three digits can be any digit from 0 to 9. All letters and digits can be repeated.2.Find the number of possible unique positions for all five light switches.pared with the number of Alberta license plates available in 1912, find the increase in the number of license plates available in 1941.In 1912, Alberta license plates consisted of four digits. Each digit could be repeated, but the first digit could not be zero. By 1941, Alberta license plates consisted of five digits. Each digit could be repeated, but the first digit could not be zero.4.Karen’s grandfather has a bank card that requires a four-digit Personal Identification Number (PIN). All digits from 0 to 9 can be used.How many PINs are possible if the digits can repeat?How many PINs are possible if the digits cannot repeat?5.Eight teenagers are lined up to buy tickets to a rock concert. How many possible arrangements of teenagers are there? Leave your answer in factorial notation.LESSON 1 SUMMARYIn this lesson you learned how to determine the total number of outcomes in a sample space by using tree diagrams, outcome charts, and the fundamental counting principle. You also learned a short-cut notation called factorial notation. By using both the fundamental counting principle and factorial notation, you have seen that there are many ways to solve problems that involve arranging items and finding the possible number of arrangements.The fundamental counting principle can be used to determine the number of possible arrangements of items. If one task can be performed in a ways, a second task in b ways, and a third task in c ways, the number of ways to calculate the possible arrangements of all three tasks is a × b × c.Factorial notation can be used as an abbreviation for products of successive integers.when n is a natural number.Lesson 2: Permutations1.Manny needs to create a four-digit password for his cell phone. If no digits can be repeated, in how many ways can this task be done?2.How many three-letter arrangements can be made from the letters in VERTICAL if no letter can be used more than once and each arrangement is made up of a vowel between two consonants?3.In a gymnastics club, 10 gymnasts practice forming a human pyramid. The pyramid uses only 6 gymnasts at any one time. There are 3 gymnasts in the bottom row, 2 in the second row, and 1 at the top.If any gymnast can fill any position in the pyramid, how many different arrangements of gymnasts are there?4.Bart wants to plant 8 trees in a row along his fence. He has been given 4 birches, 1 spruce, 1 poplar, 1 willow, and 1 elm. If the 4 birches are identical, how many possible arrangements of trees are there? 5.Solve for r.5Pr = 606.Simplify .7.Solve the equation for n.8.Nine athletes have made it to the zone cross-country running finals. Based on her previous times, Hariette is considered the favourite to win. If she does win, in how many different orders can the other racers finish?9.A local hockey club is setting up an online system to register its players.Seven-character passwords will be assigned to each player, all starting with the letter S to indicate the home club. The other characters in the password are made up of the numerals 4 through 9, and no digit can be repeated. How many passwords can the hockey club create?10.Explain why n must be greater than or equal to r in the expression LESSON 2 SUMMARYIn this lesson you learned how to find the number of permutations of objects in several ways. These ways included making a list, drawing a tree diagram, drawing blanks, using the fundamental counting principle, and using the permutation formula, nPr. You learned that there are fewer permutations if there are identical items and that some methods of determining the number of permutations are better and more efficient than others depending upon the question posed.Permutations are used when the order of the objects is important.nPr, where 0 ≤ r ≤ nThe number of permutations of n objects when a objects are identical, b objects are identical, c objects are identical, and so on isIn Lesson 3 you will learn about the number of choices of elements when order does not matter.Lesson 3: CombinationsA car dealership is promoting a particular model of car that has 8 optional features available. Each optional feature can be purchased separately.1.How many different packages of 3 optional features are possible for this model of the car?2.Explain why the number of different packages of 5 optional features is the same as the number of different packages of 3 optional features.3.Another model of car has n different optional features available. When 2 optional features are chosen for this model of car, there are 45 different packages available. Determine algebraically the number of optional features, n, that are available for this model of car.4.How many different 5-card hands with at least 3 hearts can be dealt from a standard deck of 52 cards? Hint: See “Did You Know?” on page 535 of the textbook if you are not familiar with a standard deck of cards.5.How many 6-member committees can be formed from 8 girls and 9 boys if the following are true?There are no conditions.There must be exactly 3 girls.There must be at least 4 boys.6.Create a question that involves both a permutation and a combination. Solve both showing all your steps.LESSON 3 SUMMARYCombinations can be used to solve counting problems where the order is not important. A combination can be represented by nCr or where n is the size of the group from which the objects are taken and r is the number of objects that are taken at a time. Many problems will require you to calculate multiple combinations and then apply the fundamental counting binations are used when the order of the objects is not important. The following formula can be used to calculate a combination where 0 ≤ n ≤ r Lesson 4: The Binomial Theorem1.Write an explanation for a student in Mathematics 30-1 about how to expand (a b)3. Include at least two methods of expanding the power of the binomial. 2.Use the binomial theorem to expand (x2 2y)6.3.Determine the exact value of the 12th term in the expansion of 4.Determine the term containing x7 in the expansion of LESSON 4 SUMMARYExpanding a binomial by multiplying is a very tedious process for any exponent larger than 3. It is possible to use patterns to help simplify this process. The exponents of the expansion follow an ascending and descending pattern, while the coefficients of the expansion can be found using Pascal’s triangle.Determining the values of Pascal’s triangle for large exponents would take a long time, so combinations are used to determine entries of Pascal’s triangle to simplify this process. The generalized result is the binomial theorem.The binomial theorem is used to expand a binomial using a pattern that involves combinations. The generalized binomial theorem isAny term of the expansion can be determined using the formula tk + 1 = nCk (x)n ? k (y)kMODULE 8 – PERMUTATIONS, COMBINATIONS & BINOMIAL THEOREM SUMMATIVE ASSIGNMENTComplete the following questions from your text book. Show steps completely and clearly, as marks are assigned for mathematical literacy and communication. Always use graph paper, rulers, and pencils as necessary. Attach securely to this booklet before you hand everything in.Text: Pre-Calculus 12 - Chapter 11Section 11.1: Pages: 524 to 527 #2a,c, 6, 7a,c,d, 10Section 11.2: Pages: 534 to 536 #3a, 5, 6a,c,d, 10Section 11.3: Pages: 542 to 545 #2a,c, 3a,c, 5a, 6a, 7a,d,e, 12, 15, 17a,c, 18Module 8 is now complete. Once you have received your corrected work, review your instructor’s comments and prepare for your module eight test. Congratulations!!Once all eight modules and module tests are completed, you are done the course!!! Now you need to prepare for your Diploma Examination the date set provincially and for your final exam to be written as soon as possible. Discuss these dates with Mrs. Martel! ................
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