Combinatorics, counting, probability theory

Question 1

What is the multiplication Principle?

Answer 1

If n1 ways of doing T1 and n2 –||– T2.
Then n1*n2 ways of preforming both together.

Question 2

• n elements
• Choose r elements
  What is a list of ordered elements with repetitions called? and what is the formula?

Answer 2

Sequence and n^r

Question 3

• n elements
• Choose r elements
  What is a list of ordered elements without repetitions called? and what is the formula?

Answer 3

Permutation and the formula is n!/(n-r)!

We can also write nPr which is translated to: “Number of permutations of n objects taken r at a time”

Question 4

• n elements
• Choose r elements
  What is a list of unordered elements with repetitions called? and what is the formula?

Answer 4

Multiset and the formula is (n+r-1 /r) (Not a division, but a vector)

Question 5

• n elements
• Choose r elements
  What is a list of ordered elements without repetitions called? and what is the formula?

Answer 5

Set and the formula is n!/r! * (n-r)!
Can also be written as (n r) as a vector which means “Number of combinations of n objects taken r at a time” or “n-choose-r”

Question 6

How many different 5-card hands can you choose from a normal deck of 52 cards?

Answer 6

This is a set and we have n=52 elements and we have to choose r=5 cards. Then it’s 52-choose-5 or (52 5) or 52! / 5! 47! =
5251504948 / 5432*1 = 2,598,960

Question 7

What is the formula for a Distinguishable premutation?
For a collection of n objects where the first object appears k1 times the next k2 times and so on:

Answer 7

n! / k1! * k2! …kt!

Question 8

What is the Distinguishable premutation for Mississippi?

Answer 8

Formula: n! / k1! * k2! …kt! = 11! / 1!4!4!*2! = 34,650

Question 9

What is the possible number of outcomes when rolling 3 indistinguishable dice?

Answer 9

We have a 3-element multiset, with n=6 (because we can roll from 1-6) and r=3.
from the formula (n+r-1 over r) = 8!/3! * 5! =
876/321 =56

Question 10

What defines the pigeonhole principle?

Answer 10

If m pigeons are put into n pigeonholes for m>n, then at least on pigeonhole contains at least two pigeons.

Question 11

Probability Theory:
When all events are equally likely what formula can you use?

Answer 11

We can use the following |E| / |A| =
|number of outcomes in E| / |Total number of outcomes|

Question 12

What is the probability of drawing 3 kings from a deck of cards when drawing 3 cards?

Answer 12

Here E is the 4 kings in the deck so E =4 and our A is all the possible ways for us to pick 3 out of the 52 cards. 52-choose-3. So the probability is P(E) = 4 / 52-choose-3 = 4/22100 = 1/5525

Question 13

The three different set operations:
Union, Intersection, Complement:
What does the three operations do on a an event E and F.

Answer 13

The union of E (union) F = {a in A | a (in) E or a (in) F}

The intersection of E (interest) F = {a (in) A | a (in) E and a (in) F}

The Complement of E (skrives E^c) = {a (in) A | a (not in) E}

Question 14

If we want to find the precent of an event E, how can we find it?

Answer 14

P(E) = 1 - P(E)

Question 15

If we want to find the percentage of the union between to events E (union) F, how do we find it?

Answer 15

P(E (union) F) = P(E) + P(F) - P(E (intersection) F)

Question 16

Probability of rolling odd number(E) or prime number(F) (With a fair dice)

Answer 16

E = {1,3,5} P(E) = 3/6
F ={2,3,5} P(F) = 3/6
E (union) F = {1,2,3,5} P(E (union) F) 4/6
E (intersection) F = {3,5} P(E (intersection) F) = 1/3

2/3 = 1/2 + 1/2 - 1/3