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365 Data Science - Probability > Combinatorics > Flashcards

Combinatorics Flashcards

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1
Q

What does combinatorics deals with?

A

combinations of objects from a specific, finite set

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2
Q

What Restrictions can be formed to apply to combinations?

A

Repetition
Order
Or other criterion

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3
Q

What are the 3 integral parts of Combinatorics?

A
  1. Permutations
  2. Variations
  3. Combinations

we use these to determine the number of favorable outcomes OR
the number of all elements in a sample space

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4
Q

What do Permutations represent?

A

Permutations represent the number of different possible ways we can arrange a set of elements

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5
Q

What is the formula for Permutations?

A

n!

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6
Q

How are Permutations denoted?

A

P()

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7
Q

Going back to the Formula 1 example from the lecture, in how many ways can the 20 drivers finish a race?

A

20!

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8
Q

When we determine the available options for every position in a permutation, can we start with the middle element?

A

Yes, choosing the order is completely up to us.

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9
Q

What are factorials denoted as?

A

!

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10
Q

What does n! represent?

A

The product of the natural numbers from 1 to n

ie. n! = 1 x 2 x 3 x …. x n

3! = 1 x 2 x 3 = 6

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11
Q

What is one odd characteristic of factorials?

A

Negative numbers don’t have a factorial

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12
Q

What are some important properties of factorials?

A

n! = (n-1)! x n

(n + 1)1 = n! x (n +1)

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13
Q

What is 5! Equal to?

A

120

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14
Q

Which of the values below is NOT the equivalent to “6!” ?
A) 123456
B) 7!/7
C) 4!5*6
D) 8!/8

A

8!/8

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15
Q

Which of the following is equivalent to 5!/3! ?
A) 45/6
B) 123
C) 45
D) 1*2

A

C) 4*5

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16
Q

What is the formula for Variations without Repetition?

A

V = n! / (n - p)!

The number of variations without repetition when arranging ‘p’ elements out of a total of ‘n’ elements.

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17
Q

What do Variations express?

A

The total number of ways we can pick and arrange some elements of a given set

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18
Q

What is ‘n’ in probability theory?

A

n = the total number of elements we have available

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19
Q

What is ‘p’ ?

A

p = the number of positions we need to fill

When arranging ‘p’ elements out of a total of ‘n’

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20
Q

What is the Notation and Formula for calculating Variations with Repetition?

A

V = n to the power of p

V bar, n, p = n to the p

n = the total number of elements we have available
p = the number of positions we need to fill

The number of variations with repetition when picking p-many elements out of n elements, is equal to n to the power of p

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21
Q

What is the formula for Variations with Repeating values?

A

n to the power of p

22
Q

When do we use Variations instead of Permutations?

A

When we are not arranging all elements in the sample space.

23
Q

Going back to the combinations lock example from last lecture, imagine the correct code consisted of a single 4-letter word. Remember that we can only use the letters A, B and C. How many possible passcodes are there?

3^4

4^3

2^3

26^4

A

n to p
or
3^4

24
Q

How do we calculate Variations for events without repetition?

A

We cannot use the same element twice

example element: person in running schedule

25
Q

What is the Notation and Formula for Variations without repetition?

A

n! / (n - p)!

26
Q

Going back to the relay example from the lecture, what if instead of 5 people on the team, we had 7. We would still have to choose 4 of them and arrange them in what order to run, but in how many ways can we accomplish that?

7!

4!

7!/4!

7!/3!

A

7!/3!

27
Q

Going back to the relay example from the lecture, what if instead of 5 people on the team, we had 7. Furthermore, this time we also need to pick a reserve player to be on stand-by. In how many ways can we accomplish that?
(4 positions to fulfill in the race)

7!/2!
7!/5!
7!/4!
7!/3!

A

7!/2!

28
Q

Imagine you were asked to order a 3-tier cake for your best friend’ s wedding. They asked you to get a cake with a variety of flavors, so you decide to order different filling for each tier. The pastry shop you contacted offers 5 different fillings, so you want to know how many distinct options you have for the cake.

5!/3!
5!/2!
2!/5!
3!/5!

A

5!/2!

29
Q

What do Combinations represent?

A

the number of different ways we can pick certain elements of a set

30
Q

Do Variations take into account double counting elements?

A

No

31
Q

Do Combinations take into account double counting elements?

A

Yes

32
Q
A

All the different permutations of a single combination are different variations

33
Q

In Combinations is the order relevent?

A

No

34
Q

Example: Alex, Sarah, Dave
How many variations can you have?
How many combinations?

A

6 variations - why? Pn = n!
1 Combination -

Any of the 6 permutations we showed is a different variation, but NOT a different combination

35
Q
A

6 Permutations = P(3) = 6
120 Combinations = C10,3 = 120
720 Variations = V10,3

36
Q

What is the formula for Combinations without Repetition ?

A

What is the number of combinations for choosing p-many elements out of a sample space of n elements?

C = V / P

C = number of variations / number of permutations

C = n! / p! * (n - p)!

37
Q

Imagine you are on a trip to Paris and decide to try some of their famous macaroons. The bakery you go to offers a different size “variety” packs, where you get to choose 3, 5 or 8 macaroons. The only requirement is that they all be different flavors. How many different 3-macaroon packs can you get, considering there are 8 distinct flavors.
8!/(3!5!)
8!/3!
8!/5!
3!/8!

A

8!/(3!5!)

38
Q

Now imagine you want to get the medium pack which contains 5 macaroons instead of 3. How many different possible packs can you make?
8!/5!
8!/(5!3!)
8!/3!
5!/8!

A

8!/(5!3!)

39
Q

Now imagine the same scenario but this time you want the large box of 8 macaroons. How many different variety packs can you get?

Undefined because we get 0! in the denominator of the formula.
8!(3!5!)
0
1

A

1

40
Q

What are Symmetry of Combinations?

A

We can pick p-many elements in as many ways as we can pick n minus p elements

41
Q

What is the formula for Combinations with repetition?

A

C bar = (n + p -1)! / (n -1)! * p!

42
Q

Can picking more element lead to fewer combinations?

A

Yes

43
Q

You are going on a pick nick. You have 6 pieces of fruit. However, our basket only carries 4. How many combinations do you have?

A

15

C 6,4 = 15

44
Q

Picking 4 fruits out of 6. What are the two possible ways of choosing this combination?

A

Picking 4 fruits out of 6 is the same as choosing 2 fruits that will be left out

We can pick p-many elements in as many ways as we can pick n minus p elements

C n = C n
p n - p

45
Q

When do we use Symmetry of Combinations?

A

To simplify calculations

46
Q

What are Combinations of Separate Sample Spaces?

A

A combination can be a mixture of different small, individual events

47
Q

You are developing FB ads. You have 3 different Post Descriptions, 5 thumbnails, 3 headings and 2 button options. How many different ads would you have to generate to make sure you’ve tried all possabilities?

A

90

3 x 5 x 3 x 2

48
Q

How many different burger menus can we order from a McDonald’s Restaurant if we have a choice of 8 burgers, 3 sizes of fries and 5 different drinks, assuming a menu consist of a burger, some fries and a drink?

835
8! 3! 5!
8!/(3!5!)
8*5^3

A

835

49
Q

We use Permutations with Variations when…?

A

the order is crucial

50
Q

What are the two Types of Variations and Combinations?

A

Ones with and without repetition

51
Q

What are factorials?

A

the product of an integer and all the integers below it;

52
Q

What does 0! equal?

A

0! = 1

365 Data Science - Probability (6 decks)

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