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Discrete Math Problems


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#1 Mercenary(FH)

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Posted 19 October 2009 - 04:58 PM

(yet again I know, but no book so......)
ill try to fill in my guesses my answers will be bolded

Problem 1 (2 pts.): Let S = {a, b, c}.
(a) List all the 2-permutations of S.
(b) List all the 2-combinations of S.

(a) {a,b} {a,c} {b,c} ?
(b) {a,b} {b,a} {b,c} {c,b} ? (does {b,b} etc.. count in any of them?)

Problem 2 (3 pts.): How many bit strings of length 12 contain
(a) exactly five 1’s?
(b) at most five 1’s?
© at least five 1’s?

(a) C(12,5) (12 choose 5)
(b) 1+12+(12!/2!10!)+(12!/3!9!)+(12!/4!8!)+(12!/5!7!)
© 2^12 - (1+12+ C(12,2) + C(12,3) + C(12,4) )




Problem 3 (3 pts.): A group contains k men and k women.
(a) How many ways are there to arrange them in a row?
(b) How many ways are there to arrange them in a row if the men and
women alternate?

(a) no idea? but i'd guess just 2*K!?
(b)  K! * K! * 2 = 2(K!)^2


Problem 4 (3 pts.): A club has 25 members.
(a) How many ways are there to choose a committee consisting of 4
members?
(b) How many ways are there to choose a president, vice president,
secretary and treasurer if no person can hold more than 1 position?


(a) C(25,4) = 25!/21!4!=12650
(b)  P(25, 4) = 25× 24×23×22 = 303600



Extra Problem (3 pts.) : How many ways are there for a horse race
with three horses to finish if ties are possible? (NOTE: Two or three horses
may tie.)


C(3; 3) = 1 where three horses tie.
(b)C(3; 2)C(1; 1) = 3 where two horses tie and one horse ranks second.
©C(3; 1)C(2; 2) + C(3; 1)C(2; 1)C(1; 1) = 9 where for one case one horse
ranks ¯rst and the other two horses tie. Or one horse ranks ¯rst and one
horse ranks second and one horse ranks third.


so 9+3+1 = 13


How do my answers look? some I guess, some I used interweb resources.

thx

Edited by Mercenary(FH), 19 October 2009 - 05:01 PM.


#2 omfgzila

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Posted 01 January 2010 - 10:04 AM

QUOTE(Mercenary(FH) @ Oct 20 2009, 01:58 AM)  

(yet again I know, but no book so......)
ill try to fill in my guesses my answers will be bolded

Problem 1 (2 pts.): Let S = {a, b, c}.
(a) List all the 2-permutations of S.
(b) List all the 2-combinations of S.

(a) {a,b} {a,c} {b,c} ?
(b) {a,b} {b,a} {b,c} {c,b} ? (does {b,b} etc.. count in any of them?)

Problem 2 (3 pts.): How many bit strings of length 12 contain
(a) exactly five 1’s?
(b) at most five 1’s?
© at least five 1’s?

(a) C(12,5) (12 choose 5)
(b) 1+12+(12!/2!10!)+(12!/3!9!)+(12!/4!8!)+(12!/5!7!)
© 2^12 - (1+12+ C(12,2) + C(12,3) + C(12,4) )

Problem 3 (3 pts.): A group contains k men and k women.
(a) How many ways are there to arrange them in a row?
(b) How many ways are there to arrange them in a row if the men and
women alternate?

(a) no idea? but i'd guess just 2*K!?
(b)  K! * K! * 2 = 2(K!)^2


Problem 4 (3 pts.): A club has 25 members.
(a) How many ways are there to choose a committee consisting of 4
members?
(b) How many ways are there to choose a president, vice president,
secretary and treasurer if no person can hold more than 1 position?


(a) C(25,4) = 25!/21!4!=12650
(b)  P(25, 4) = 25× 24×23×22 = 303600

Extra Problem (3 pts.) : How many ways are there for a horse race
with three horses to finish if ties are possible? (NOTE: Two or three horses
may tie.)


C(3; 3) = 1 where three horses tie.
(b)C(3; 2)C(1; 1) = 3 where two horses tie and one horse ranks second.
©C(3; 1)C(2; 2) + C(3; 1)C(2; 1)C(1; 1) = 9 where for one case one horse
ranks ¯rst and the other two horses tie. Or one horse ranks ¯rst and one
horse ranks second and one horse ranks third.
so 9+3+1 = 13


How do my answers look? some I guess, some I used interweb resources.

thx

question with horses: you use the formula (n+k-1 / k) so it ends up being 5 nCr 3 what is 10.

club problem is okay

men and women: a) is (2K)! not 2K!; b)correct


QUOTE

Problem 1 (2 pts.): Let S = {a, b, c}.
(a) List all the 2-permutations of S.
(b) List all the 2-combinations of S.

You mixed them, there are six P and three C. P is where the row matters so there are automatically more of them because is it c,b or b,c matters, unlike in combinations. (get this example into your head: permutations-race(row matters), combination-cards(row doesnt matter))

as for problem two, my english is kinda bad so I didn't understand what you mean






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