Solving for permutations and combinations

Question
1) how many different ways can a teacher select 2 books from a possible 17 books?
2) how many different ways can be made from a test bank of 20 questions if the test consists of 5 questions?
3)how many different ways can 4 tickets be selected from 50 tickets if each tickets wins a different prize?



Solution
These are examples of permutations and combinations problems.
The problems can be solved in two ways depending on the assumptions.

Assumption 1 below is an example of a combination.
For example how many different committees of three students can be chosen from a group of 10?
Therefore,
10!
---
3!7!
which is equal to
10.9.8
------
3.2.1
to get 120.
Assumption 2 below is an example of a permutation.
For example, the permutation of three letters from the set a,b,c are 6 which are enumerated below
abc
bac
cab
acb
bca
cba
Which can also be computed by
3!
--
0!
which is equal to
6
-
1
to get 6.
Assumption 1: The order of the arrangements or combinations DOES NOT matter
Question 1
17!
---
2!
which is
17.16
-----
2.1
to arrive at 136
Question 2
20!
----
5!
which is
20.19.18.17.16
---------------
5.4.3.2.1
to arrive at 15,504
Question 3
50!
----
4!
which is
50.49.48.47
------------
4.3.2.1
to arrive at 230,300

Assumption 2: The order of the arrangements or combinations DOES matter
Question 1
17!
---
15!
which is 17.16
to arrive at 272.
Question 2
20!
----
15!
which is 20.19.18.17.16
to arrive at 1,860,480

Question 3
50!
----
46!
which is 50.49.48.47
to arrive at 5,527,200
You may also want to refer to
How many possible combinations can pennies be distributed?
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