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CSE 103 > Lec 4: Combinatorics 2 > Flashcards

Lec 4: Combinatorics 2 Flashcards

1
Q

What is the combinatorial function and its boundary conditions?

A
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2
Q

What is the binomial expansion of (a+b)2 and (a+b)3?

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3
Q

What is the binomial expansion equation sing the conbination function?

A
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4
Q

What is inductive computation of C(n,r) where r is the size of a subset from a set of size n? What is the general formula?

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5
Q

What is Pascal’s Triangle use for?

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6
Q

What is the pigeon-hole principle?

A

if n items are put into m containers, with n > m, then at least one container must contain more than one item.

For natural numbers k and m, if n = km + 1 objects are distributed among m sets, then the pigeonhole principle asserts that at least one of the sets will contain at least k+1 objects.

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7
Q
A

3*(6-1)+1 = 16

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8
Q

What is the birthday paradox?

A
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9
Q

How many people do you need in the room so that at least two of them have the same birthday?

A

Assume all days have the same probability (1/365)

K = the number of people in the room

We want to calculate P(A) for the event

A = {K birthdays such that at least two are the same}

P(A) = | A | / | outcome space |

Easier to think of it as the 1 - the complement, no two people have the same birthday

| Ac | = 365! / (365 - K)!

P(A) = 1 - P(Ac) = 1 - P(k,365)/365k

outcome space | = 365k

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10
Q

How many strings contain 3 letters and two digits? (digits and letters can repeat and there is no restrictions on their order)

A
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11
Q

What is the number of strings that start with a digit followed by 4 letters, followed by 2 digits?

A

Answer: this is a product set:

10*26*26*26*26*10*10 = 26^4*10^3

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12
Q

What is the probability that a random word of length 4 with distinct letters has the letters in increasing alphabetic order?

A
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13
Q

How many ways to sit 3 out of 7 kids on a marry-go-round with three identical seats?

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14
Q

What is stars and bars?

A
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15
Q

How many ways to divide 10 cookies among three children? (the cookies are identical and cannot be broken)

A
  • stars and bars problem

-

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16
Q

How many ways to split 10 cookies among 3 kids if each kid has to get at least 2 cookies?

A
  • stars and bars problem
    1. First, give each kid 2 cookies, 4 cookies are left.
    2. Second, divide the remaining cookies among the 3 kids.

C(4+3-1,3-1) = C(4+3-1,4)

17
Q

You have a shelf with 24 books on it. You can pick any 3 books but no two picked books can be right next to each other. How many choices do you have?

A
18
Q

If you choose 3 out of 24 books at random, what is the probability that at least 2 would be next to each other?

A
19
Q

[WebWork 2.2.a]

A

{0}

20
Q

[WebWork 2.2.b]

A

{1}

21
Q

[WebWork 2.2.c]

A

{0,2,3}

22
Q

[WebWork 2.3]

A
23
Q

What is De Morgan’s Law for Sets?

A
24
Q

[WebWork 2.4]

A
25
Q

[WebWork 2.5.a] For lunch, I can eat only one sandwich and only one piece of fruit. My lunch must includeboth sandwich and fruit. For sandwich, I can choose between veggie, chicken, and beef sandwiches. For fruit, I can choose between an orange and an apple. How many different lunch combinations can I have?

A

3*2

26
Q

[WebWork 2.5.b] A particular brand of shirt comes in 13 colors, has a male version and a female version, and comes in 5 sizes for each sex. How many different types of this shirt are made?

A

13*2*5

27
Q

[WebWork 2.6.a] How many strings of letters can be formed by rearranging the letters “CAB”? The order matters in the rearrangement. For example, “BCA” is one rearrangement of “CAB”. The original arrangement “CAB” should be counted.

A

3!

28
Q

[WebWork 2.6.b] A pianist plans to play 6 different pieces at a recital. how many ways can she arrange these pieces in the program? You should related this problem with the previous problem by taking pieces as letters.

A

6!

29
Q

[WebWork 2.8]

A
30
Q

[WebWork 2.9.a] How many ways are there of picking one digit (from 0-9) followed by one letter (from A-Z)? Note the order of digit and letter is fixed. For example, ‘1Z’ is a valid outcome, but ‘Z1’ is not.

A

10*26

31
Q

[WebWork 2.9.b] Standard automobile license plates in a country display 2 numbers, followed by 3 letters, followed by 2 numbers. For example, Note the order matters here. For example, a license plate displays 2 numbers, followed by 1 letter, followed by 1 number, can be 12A1, which is different from 21A1. How many different standard plates are possible in this system? (Assume repetition of letters and numbers is allowed.)

A
32
Q

[WebWork 2.10.a]

Suppose we choose two different numbers, each from 1 to 5. If the order of the numbers is NOT important, how many different choices can we pick?

A

C(5,2)

33
Q

[WebWork 2.10.b] In the “6/55” lottery game, a player picks six numbers from 1 to 55. How many different choices does the player have if repetition is NOT allowed? Note again that the order of the numbers is NOT important.

A

C(55,6)

34
Q

What is uniform distribution over finite outcome spaces?

A

A uniform distribution over a set of size n is one which assigns to each element the probability 1/n. For example the probability that the outcome of a dice throw is 3, is equal to 1/6. The other 5 possible outcome {1,2,4,5,6} also have a probability of 1/6 each.

35
Q
A
36
Q

[WebWork 2.11] Suppose we have 16 cards, numbered 1,…,16. Suppose that we randomly pick a card from the 16 cards. So, the outcome space is {1, 2, 3, … , 16} and the probability that we get any numbered card is 1/16. Let A be the event that number of the card that we picked is smaller or equal to 15. What is the probability of the event A?

A
37
Q

[WebWork 2.12]

A
38
Q

[WebWork 2.13]

A

What is the probability of getting exactly 6 heads?

(11! / (6!5!)) / 2^11

39
Q

[WebWork 2.14]

A

CSE 103 (10 decks)

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