MTH 245: Mathematics for Management, Life, and Social Sciences

Transcription

1 1/1 MTH 245: Mathematics for Management, Life, and Social Sciences Sections 5.5 and 5.6. Part 1

2 Permutation and combinations. Further counting techniques 2/1 Given a set of n distinguishable objects. Definition 1 permutation of n objects taken k at a time is an arrangement of k of the n objects in an specific order. Definition 2 combination of n objects taken k at a time is a selection of k objects among the n, with order disregarded. Notation P(n,k) = the number of permutations of n objects taken k at a time. C(n,k) = the number of combinations of n objects taken k at a time.

3 Permutation and combinations. Further counting techniques Further counting techniques. 3/1 Permutation formula For example, for k = 1,2 and 3 P(n, 1) = n P(n,2) = n(n 1) P(n,3) = n(n 1)(n 2) Formula: Permutation formula The number of permutations of n objects taken k at a time is P(n,k) = n! (n k)! = n(n 1)(n 2) (n k + 1).

4 Permutation and combinations. Further counting techniques Further counting techniques. 4/1 Combination formula The number of combinations of n objects taken k at a time is Formula: Combination formula C(n,k) = P(n,k) k! = n! k!(n k)! = n(n 1) (n k + 1) k(k 1) 1 Note: 0! = 1 and 1! = 1

5 Permutation and combinations. Further counting techniques Further counting techniques. 5/1 Permutations and combinations Consider the set {a,b,c,d} and consider all subsets with three elements: Subsets (combinations) {a,b,c} {a,b,d} {a,c,d} {b,c,d} Ordering (permutations) abc, acb, bac, bca, cab, cba abd, adb, bad, bda, dab, dba acd, adc, cad, cda, dac, dca cbd, cdb, bcd, bdc, dcb, dbc There are 6 permutations for each combinations. makes sense... C(4,3) = P(4,3) 3!

6 Permutation and combinations. Further counting techniques Further counting techniques. 6/1 Examples Example 1 Compute C(9,4). What does C(9,4) represent? Example 2 Compute P(9,4). What does P(9,4) represent?

7 Permutation and combinations. Further counting techniques Further counting techniques. 7/1 More examples... Example 3 SIAM Oregon State Student chapter needs a president, a vicepresident, a secretary, a treasurer and a web-master for this year. The chapter has 30 members with the right to nominate candidates for each position. In how many ways can president, vice-president, secretary, treasurer and web-master be chosen among the chapter members?

8 Permutation and combinations. Further counting techniques Further counting techniques. 8/1 Example 4 Given the set {a,b,c} 1 Count the number of arrangements of the sets of the elements of the set {a,b,c}, taken two at a time, without allowing repetitions. The order of the arrangement matters. 2 Count the number of arrangements of the sets of the elements of the set {a,b,c}, taken two at a time, without allowing repetitions. The order of the arrangement does NOT matter.

9 Permutation and combinations. Further counting techniques Further counting techniques. 9/1 Page 218, #46 Example 5 A nautical signal consists of three flags arranged vertically on a flagpole. If a sailor has six different flags, how many different signals are possible?

10 Permutation and combinations. Further counting techniques Further counting techniques. 10/1 Airport codes (Page 217, #22) Example 6 Find the number of different airport codes in which each code consists of three letters. With repeats With no repeats

11 Permutation and combinations. Further counting techniques Further counting techniques. 11/1 Example 7 ow many different words can be formed from the letters of the word COMMITTEE. We are counting list of words not necessarily found in the dictionary. It is NOT just P(9,9). Why? Note the following: COM 1 M 2 IT 1 T 2 E 1 E 2 looks like COM 2 M 1 IT 1 T 2 E 1 E 2. Anything else? Exercise: Solve the same problem with the word BELLE

12 Permutation and combinations. Further counting techniques Further counting techniques. 12/1 Choose a password Example 8 In how many different ways can we choose an eight character password with letters and numbers 1, R, 2, 1, E, R, 3, R.