Data Management Page 1 of 12 Permutations Extra Problems (solutions)

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Leon Miller
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1 Data Management Page of. a) How many -digit numbers can be formed using the digits 0,,,,, if no digits may be repeated in the number? remaining digits, including zero not zero {,,,, } b) How many of the above numbers will be odd? odd digits, {,, } not zero, not remaining digits, including zero c) How many of the above numbers will be even? i) last digit is zero ii) last digit is or zero {, } remaining digits not zero, not remaining digits (i)+(ii) = 0 + = possible even numbers

2 Data Management Page of. How many odd, -digit numbers, all of the digits different, may be formed from the digits 0 to 9 inclusive, if there must be a in the number? a) Solve using an indirect approach. (i) all odd -digit numbers -(ii) all odd -digit numbers without a i) Using {0,,,,,,,,, 9} odd; {,,,, 9} remaining digits, including zero not zero; not ii) Using {0,,,,,,,, 9} odd; {,,, 9} remaining digits, including zero not zero; not (i)-(ii) = 0 = 0

3 Data Management Page of b) ive an alternate solution to this problem which uses the direct approach. Using {0,,,,,,,,, 9}, we have cases, moving the digit along. i ) {} odd; {,,, 9} remaining digits, including zero ii) {} odd; {,,, 9} not zero; not and not remaining digits, including zero iii) {} odd; {,,, 9} not zero; not and not remaining digits, including zero iv ) {} not zero; not remaining digits, including zero (i) +(ii)+(iii)+(iv) = = 0

4 Data Management Page of. How many even -digit numbers with no digits repeated do not contain: a) either a or a? i) the last digit is zero Using {0,,,,,,, 9} {0} remaining digits ii) the last digit is even, but not zero {,,, } not zero; and not remaining digits (i) +(ii) = 0+0 = 0 b) either a or a? Using {0,,,,,,, 9} i) the last digit is zero remaining digits {0} ii) the last digit is even, but not zero {,, } remaining digits not zero; and not (i) +(ii) = 0+0 = 000

5 Data Management Page of. In how many natural numbers less than does at least one of the digits 0,,,,,,, occur? (Hint: Use an indirect approach.) (all #s) (no 0,,,,,,, ) i) all natural numbers < numbers ii) all numbers consisting of {, 9} (-digit; -digit; -digit; -digit cases) = 0 (i) (ii) = = 999 possible numbers.

6 Data Management Page of. a) Find the number of even -digit numbers using digits from to with no duplication. Using {,,,,,, }; even {,,, } even remaining digits b) How many of the above numbers contain a? (answer from (a)) (even #s without ) remaining digits even c) How many of the above numbers contain a and if the digit is in the number, it must come last? i) is the last digit & there are cases moving, as shown {} {} {} {} {} {} {} {} ii) is not included & there are cases moving, as shown Using {,,,,, } (i) +(ii) = ( )! + P = cases P cases = 0 {} {} {} {} {} {, } {, } {, } {, }

7 Data Management Page of. a) How many different signals can be made by arranging signal flags on a vertical mast if there are flags available each of a different colour? A signal must include at least one flag. different colours, {A,, C, D, E} cases: flag; flag; flags; flags; flag signals P + P + P + P = +! = b) How many different signals can be made using up to flags if you have red flags and blue flags? different colours, {A, A, A,, } cases: flag; flag; flags; flags; flag signals. a) In how many ways can boys and girls stand in a row if no boy is beside another boy and no girl is beside another girl? (! ) cases = 00 or b) If you removed the restriction from part a) that no girl is beside another girl but kept the restriction that no two boys can be beside one another, in how many ways could the 0 children stand in a row? Listing cases: (!) x cases = 00 alternating s together Therefore, cases.

8 Data Management Page of. Find the number of ways of arranging the letters of the word TENNESSEE a) if there are no restrictions. E 9!!!! N = 0 S TENS ENS E E b) if the first letter must be E and the last letter must be an N. E remaining letters N TENS E S E c) if all of the E's are always together.! = 0!! N S { EEEE, T, N, N, S, S} d) if neither of the first two letters is an E. (all arrangements) (first letters must be Es) E E remaining letters TENS ENS!!!! = 0 N S

9 Data Management Page 9 of 9. Four married couples are to be seated around a circular table. The hostess wants to know how many ways she can arrange the couples around the table a) if there are no restrictions? 0! = 9! = 00 0 rotations b) if each husband must sit beside his wife? {H, W} Arrange couples in a circle:! =! = rotations {H, W} {H, W} arrangements x = 9 {H, W} Each couple could be {H, W} or {W, H} c) if each husband must sit beside his wife and men and women alternate? anyone the spouse of based on the gender of the spouse of the last couple rotations ( )( ) = another couple and their spouse d) if each husband sits opposite his wife? anyone their spouse their spouse their spouse another person another person another person the spouse rotations ( )( )( ) =

10 Data Management Page 0 of 0. How many whole numbers less than 00 can be formed using the digits,,,,, and in each of the following cases? a) No repetition is allowed. b) Repetitions are allowed. i) -digit i) -digit {,,, } remaining digits ii) -digit iii) -digit {,,,,, } {,,,,, } (i)+(ii)+(iii) = 0+0+ = {,,, } {,,,,, } ii) -digit {,,,,, } iii) -digit {,,,,, } (i)+(ii)+(iii) = ++ = c) Any digit can be used at most twice. Indirect: (all #s) (-digit triple repeats) From (b) You can count them with your fingers: {,,, } Therefore, = numbers.

11 Data Management Page of. Kurt, Yoel, Jason, Elana and Sara are studying together for a Data Management test. They decide to take a break and get a snack. Each is allowed to take one snack item. In how many different ways can the snacks be distributed if there are apples, oranges, pieces of cherry pie and blueberry muffins? snacks: {A, R, C, } Cases: i) alike ii) alike, different iii) alike, different iv) alike, alike v) alike, different vi) Kurt Yoel Jason Elana Sara alike, alike, different Hint: (i)+(ii)+(iii)+(iv)+(v)+(vi)+(vii) = = 00. Consider the digits 0 to. If repetition is allowed, how many -digit numbers: a) can be created with no restrictions? b) contain at least one digit repeated? (all -digit #s) (all distinct digits) any {0,,,,,,,, } From (a) not zero c) are divisible by? not zero remaining, including zero 9 9 Therefore, (9 ) ( P ) = not zero {0, } any {0,,,,,,,, }

12 Data Management Page of. How many permutations of the digits,,,,,,,, 9 are there in which: a) and 9 have exactly number between them? Consider 9 tied together. {} {9} _ _ cases ( before 9, or 9 before ) Therefore,! x cases x = 0 0 permutations. b) occurs before as you move from left to right? Count or, you could argue, ½ have before out of all 9! Permutations: ½ have before! 9 = 0 have before.

Discrete Structures. Fall Homework3

Discrete Structures. Fall Homework3 Discrete Structures Fall 2015 Homework3 Chapter 5 1. Section 5.1 page 329 Problems: 3,5,7,9,11,15 3. Let P(n) be the statement that 1 2 + 2 2 + +n 2 = n(n + 1)(2n + 1)/6 for the positive integer n. a)