Basic knowledge

Sample types




In a variation we draw k elements out of a set of n elements. The ordering in the resulting list of k elements is important.

Explanation
n: Number of objects in the set
k: Number of drawings = elements in the result list

If every element in the set can only be drawn once, then it is a variation without repetition (w/o-rep).
Formula:
Example:
A horse race with 10 horses takes place. How many possibilities are there for the first three places?
Solution:

Example with urn model:
A contest with 10 participants takes place. How many possibilities are there for the top 3?


If every element in the set can be drawn multiple times, then it is a variation with repetition (w-rep)
Formula:
Example:
How many possibilities are there to form a word with 5 letters from the alphabet?
Solution:

Example with urn model:
A 4-digit bike lock allows to set each digit to one of the values 0 to 9. How many possible codes exist?
In a combination we draw k elements out of a set of n elements. The ordering in the resulting subset of k elements is unimportant.

Explanation
n: Number of objects in the set
k: Number of drawings = elements in the result subset

If every element in the set can only be drawn once, then it is a combination without repetition (w/o-rep).

Formula:

Example:
Lotto 6/49: Count the number of possible outcomes!
Solution:


If every element in the set can be drawn multiple times, then it is a combination with repetition (w-rep)

Formula:

Example:
We draw 3 times out of an urn with 6 balls and return each time the ball drawn into the urn.
Solution:

Example with urn model
A fruit seller packs bags of 10 fruits from 4 different types of fruits as a special offer. How many differing bags of fruit are possible?

Solution:

n-1 is the number of separator sheets needed in order to separate in the ordered list of objects, e. g. AA|BBB|C|DDDDD the (here: 4) object classes. This is the reason for „n-1“ in the formula.

Special topics

Permutation is a special form of variation, where all n elemnts are drawn w/o repetition (n=k).

Example:
In a treasure hunt game all 5 stations have to be reached. The set of stations n is 5, thus k is 5 as well, since all stations have to be reached.
With Laplace probability, all events of an experiment have the same probability.
P(A) = Number of events which satisfy condition A = desired events
Number of all possible events possible events


Example:

How high is the probability to roll a 4 with a six-sided dice?
P({4}) = 1/6

How high ist he probability to roll a 3 or a 5 with a six-sided dice?
P({3, 5}) = 2/6

n! is read „n-factorial“.
Formula: n! = 1 ∙ 2 ∙…∙ n, n ∈

Example:
5! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ 5

Special rule:
0! = 1

For n, k ∈ ∪ {0} with k ≤ n the binomial coefficient is defined as:

(The last rule holds only for k > 0.)
Example:


A few rules
Sum rule
There are n elements with property a and m elements with property b. The a and b properties cannot be taken or occur at the same time. Therefore, there are n+m possibilities to select one object with property a or b.
Example:
In a car rental company there are 3 small cars and 7 middle-sized cars. When you need to choose one of the cars, you have in total 3+7 possibilities to choose.
Product rule
If a problem can be divided into 2 subproblems which are executed one after another, and if there are n possibilities for the 1st subproblem and m possibilities for the 2nd subproblem, then there are n*m possibilities in total.

Example:
There are 3 routes from Gummersbach to Cologne and 4 routes from Cologne to Aachen. Then there are in total 3*4 = 12 possible routes from Gummersbach to Aachen.