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Additional Problems

Permutations and Combinations. Additional Problems. Permutations and Combinations with and without Repetition. n is number of distinct classes of objects in the original bag!. - r-permutation without repetition - order matters (r distinguishable slots) - without replacement

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Additional Problems

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  1. Permutations and Combinations Additional Problems .

  2. Permutations and Combinations with and without Repetition

  3. n is number of distinct classes of objects in the original bag! - r-permutation without repetition - order matters (r distinguishable slots) - without replacement (n distinguishable objects) - r-combination without repetition - order does not matter (r indistinguishable slots) - without replacement (n distinguishable objects) n! / r! (n-r)! n! / (n-r)! - r-permutation with repetition - order matters (r distinguishable slots) - with replacement (n distinct classes of indistinguishable objects) n^r - r-combination with repetition - order does not matter (r indistinguishable slots) - with replacement (n distinct classes of indistinguishable objects) (n+r-1)! / r! (n-1)!

  4. Combinations or Permutations? • 1. In how many ways can you choose 5 out of 10 friends to invite to a dinner party? • Solution: Does the order of selection matter? • NO • If you choose friends in the order A,B,C,D,E or A,C,B,D,E the same set of 5 was chosen, so we conclude that the order of selection does not matter. • We will use the formula for combinations since we are concerned with how many subsets of size 5 we can select from a set of 10. C(10,5) =

  5. Permutations or Combinations? • How many ways can you arrange 10 books on a bookshelf that has space for only 5 books? • Does order matter? • The answer is yes since the arrangement ABCDE is a different arrangement of books than BACDE. • We will use the formula for permutations. We need to determine the number of arrangements of 10 objects taken 5 at a time so we have P(10,5) = 10(9)(8)(7)(6)=30,240

  6. Lottery problem • A certain state lottery consists of selecting a set of 6 numbers randomly from a set of 49 numbers. To win the lottery, you must select the correct set of six numbers. How many possible lottery tickets are there? • Solution. The order of the numbers is not important here as long as you have the correct set of six numbers. • To determine the total number of lottery tickets, we will use the formula for combinations and find C(49, 6), the number of combinations of 49 items taken 6 at a time. • C(49,6) = 13,983,816

  7. Examples • How many ways can a 3-person subcommittee be selected from a committee of a seven people? • The number of ways that a three-person subcommittee can be selected from a seven member committee is the number of combinations (since order is not important in selecting a subcommittee) of 7 objects 3 at a time. This is:

  8. Example (cont) • How many ways can a president, vice-president, and secretary can be chosen from a committee of 7 people? • The number of ways a president, vice-president, and secretary can be chosen from a committee of 7 people is the number of permutations (since order is important in choosing 3 people for the positions) of 7 objects 3 at a time. This is: P(7,3)

  9. Problem • From a standard 52-card deck, how many 7-card hands have exactly 5 spades and 3 hearts? • The five spades can be selected in C13,5 ways and the two hearts can be selected in C13,2 ways. Applying the Multiplication Principle, we have: Total number of hands

  10. Problem • How many ways are there to select five bills from a cash box containing $1, $2, $5, $10, $20, $50 and $100 bills, such that the bills are indistinguishable and the order in which they are selected is unimportant. (there are also at least 5 bills of each kind). • Solution • Order doesn’t matter and • Repetitions are allowed • This is like drawing colored balls with replacement. The colors correspond to the values. Since the order doesn’t matter we have: • C(7+5-1,5)=462

  11. Combinations with Repetition Approach: Place five markers in the compartments i.e., # ways to arrange five stars and six bars ... Solution: Select the positions of the 5 stars from 11 possible positions ! C(n+r-1,5)=C(7+5-1,5)=C(11,5) n=7 r=5 compartments and dividers markers

  12. exercise • A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?

  13. exercise A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Guards: Forwards: Center: Thus, the number of ways to select the starting line up is 2*10*6 = 120.

  14. Problem • A cookie shop has 4 kinds of cookies and we want to pick 6. In how many ways we can pick them? • We don’t care about the order and cookies from one kind are indistinguishable. • This is precisely the problem we saw to solve the r-combination with repetition • C(6+4-1,6)=84.

  15. Problem • In how many ways can we place 10 indistinguishable items in 8 distinguishable boxes. • This is precisely the problem we saw to solve the r-combination with repetition: • C(10+8-1,10)

  16. Problem • How many different “words” can we create by reordering SUCCESS ? • Total number of permutations is 7!. However permuting the 3 S’s does not create a new word, idem 2 C’s: • 7!/3! 2!

  17. Problem • Math teacher has 40 issues of a journal and packs them into 4 boxes,10 issues each. • a) How many ways if the boxes are numbered? •  assign boxes to issues: 40! / (10!)^4 • b) How many ways if the boxes are indistinguishable. •  There are 4! ways to label the boxes, • once we have distributed them in unlabelled boxes. • Since the number of ways to distribute them in labeled boxes is given by a) i.e..40! / (10!)^4 • we get 40! / (10!)^4 4!.

  18. Problem • How many different cross terms will we generate when we multiply out: (x1+x2+...+xm)^n ? • How many different exponents are there of the sort x1^n1 x2^n2 ... xm^nm with n1+n2+...+nm=n. • Equivalent to : how many different way are there to put n balls in m boxes: • C(n+m-1,n)

  19. Problem • In how many ways can 7 girls and 3 boys line up, if the boys must stand next to each other? • Tie-up 3 boys as one , thus it is • about ordering 8 “things” and then choosing an order for the boys inside the one “big thing”. • 8!*3!

  20. Problem • Suppose an operating system has a queue of 3 low priority and 5 high priority processes ready to run. In how many ways could these processes be ordered for execution if 2 low priority processes are not allowed to be executed back to back? • Order H in 5! Ways To get HHHHH • Then _H_H_H_H_H_ • Order L in 3! Ways • 6p3 for the 3 spots where we put the Ls. • 5!*3!* 6p3

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