Perms and Combs revision (1 Viewer)

[withoutaface]

Here's a collection of questions I copied from my discrete textbook, shouldn't be too much of a challenge for most of you, but good revision nonetheless. If you're going to post the answers, please use the [spoiler][/spoiler] tags.

1) The 10 students in a tutorial group each hand in an assignment. These assignments are then given to 3 markers. In how many ways can this be done?

2) A byte is a string of eight 1's and 0's. How many distinct bytes are there?

3) How many ways are there to seat 3 students in a 4 chair row?

4) How many six digit numbers are there that do not repeat a digit, and do not start with 0?

5) i) How many strings are there of length 3 that start with 2 digits and end with one of the 26 capital letters of the alphabet?
ii) In how many ways can 500 students seat themselves in a room containing 550 seats? (Express your answer in terms of factorials).

6) i) How many four digit numbers greater than 1000 can be formed using the digits 0, 1, 2, 3, and 4?
ii) How many four digit numbers greater than 1000, with no repeated digit, can bve formed using the digits 0, 1, 2, 3, and 4?

7) Four people are about to have a snack and there are eleven types of cake available. Each person chooses just one cake.
i) How many possibilities are there?
ii) How many possibilities are there if everyone has a different type of cake?

8) A restaurant has five entrees, seven main courses and ten desserts. In how many ways can you select two dishes on the condition they must not both be from the same part of the menu?

9) i) How many strings of 8 distinct letters can be made from the letters {a, b, c, d, e, f, g, h}?
ii) How many of the strings you found in i) do not have any of the elements of {a, b, c} next to each other?

10) You have a deck of fifty-two cards.
i) How many ways are there of choosing a hand of cards?
ii) How many of them contain the queen of hearts?
iii) In how many ways can four hands of five cards each be given to four players?
iv) In how many ways can four hands of five cards be selected from the deck?

11) Consider the set {a, b, c, d, e, f}. How many awys are there of choosing four letters from this set:
i) if no letter is chosen twice?
ii) if no repititions are allowed?

12) i) In how many ways can thirteen cards be chosen from a deck of fifty-two cards?
ii) In how many ways can fifty-two cards be divided into four lots of thirteen?

13) How many different outcomes are possible if seven identical dice are thrown?

14) Given a large supply of jelly beans of 10 different colours, how many ways are there to make up a bag of 5 jellybeans?

15) How many distinguishable arrangements are there of the characters in the words
i) hodmandod
ii) imperseverant
iii) myristicivorous
iv) indistinguishable
v) sociological
vi) Mississippi?

16) Suppose 20 people are divided into 6 different committees (labelled C1 to C6). Suppose that committee C1 is to have 3 people, C2 is to have 4 people, C3 to have 4, C4 to have 2, C5 to have 3 and C6 to have 4, how many different arrangements are there?

17) In how many ways can 15 distinct balls be placed in 4 boxes so that the first box contains 6 balls, the second box contains 3, the third box 4, and the fourth box 3?

18) Suppose a signle die is rolled 30 times, and the results are recorded in order. Suppose that the number 1 appeared 4 times, the number 2 appeared 2 times, the number 3 8 times, the number 4 5 times, the number 5 4 times and the number 6 7 times. How many possible ways are there for a string of outcomes?

 

[Jago]

thanks waf, will look at these monday arvo.

 

[Slidey]

I think question 2 should be "a byte is 8 bits (ones and zeros)", BTW.

 

[withoutaface]

Thanks, fixed.

 

[Slidey]

Thanks for these. Will do them when I cover perms and combs.

 

[Jago]

are there any answers for these?

edit: nevermind, got them

 

[withoutaface]

Spoiler

1.3^10
2. 2^8
3. 24
4. 136080
5 i)2600 ii)550!/50!
6 i) 499 ii)96
7 i) 11^4 ii)7920
8. 155
9 i) 8! ii) 5!6(C or P, not sure of the notation they use)3
10 i)52!/47!5! ii) 51!/47!4! iii) 52!/32!5!5!5!5! iv) 52!/32!5!5!5!5!4!

11 i) 15 ii)126
12 i) 52!/39!13! ii) 52!/13!13!13!13!4!
13 12C7
14 14C9
15 i)9!/2!3! ii)13!/3!2! iii)15!/2!3!2!2! iv)17!/4!2!2! v)12!/3!2!2! vi) 11!/4!2!2!
16 20!/3!4!4!2!3!4!
17 15!/5!4!3!3!
18 30!/4!2!8!5!4!7!