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?
How to recover Nokia Mobile Phone Security Code?
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How to recover Nokia Mobile Phone Security Code?

Question
How could I get a possible combination and permutation for a 5 digit code from 1-6. Locked my phone and can't remember it. I really don't wanna loose all my contacts if Nokia has to reset it.
Solution
Whew! This is a problem I can relate with. I once forgot the security code of my luggage. It is a good thing that it had only 3 digits and that I remembered the first and third numbers - so I only had to try 10 times to get the correct security code, otherwise I would have to get a locksmith.

So, how do we go about your problem? Unfortunately, it is very difficult because because there are 5 digits, however it would have been more difficult if you choose any number from 1 to 10 instead of just 1 to 6.

Given your situation, there are 7,776 possible 5 digit combination. This is computed by 6*6*6*6*6 0r 6^5! However, if you only used a number ONCE in any of the 5 digits, then your choices get narrower. In this case you will only have 720 possible combinations, which is computed by 6*5*4*3*2.

Now, if you can just remember the first 2 digits, then your choices will be scaled down to just 4*3*2 which is 24!

I hope this helps and please let me know what happens. Good luck!




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How many possible combinations can pennies be distributed?

Question
In how many ways can 3 girls divide 10 pennies if each must end up with at least one penny?




Solution
First, let us determine whether the above problem is an example of a permutation or combination problem OR the combination of both.

Combination is the various ways that the 10 pennies can be distributed to the 3 girls given that each girl must end up with at least 1 penny.
Permutation, on the other hand, is defined as the possible arrangements of the pennies.

Now, for the above problem, it does NOT matter what penny goes to each girl - what matters is that each gets at least a penny. Hence, the problem is both a combination and a permutation problem.

Second, the easiest way to solve a combination and permutation problem is by using the nPr function of your calculator.

• Turn on your calculator
• Key in '10' for the total number of pennies
  
• Press nPr
• Key in '3' for the total number of ways
• Press equal or enter key
• You get 720 possible combinations!
  

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