PERMUTATION AND COMBINATION.
FUNDAMENTAL COUNTING PRINCIPLE (FCP):
So you say you can count! Of course you may can count naturally, without even thinking about it! But can you count probabilistically, without giving it a thought?
You purchased 4 pants and 3 shirts. How many possible ways can you dress?
4*3 ways= 12 possible outfits.
In the formal definition of the (FCP) we call these items, events. Thus M occurs in m ways followed by event N that occurs n ways then event M followed by N can occur in m*n ways.
PERMUTATION is an arrangement of items in which order matters. This concept can be very elusive and so let us make a very basic and practical example.
There are three books to be shelved. Let’s call them book A,B and C for simplicity. How many possible ways you can arrange them on the shelf?
Let us count together this way:
1. ABC
2. ACB
3. BCA
4. BAC
5. CBA
6. CAB
Clearly, what books you choose to put first, second or third make a difference because order matters. These are really six distinct ways that you could have chosen to arrange them on the shelf. So there are six possible choices of arrangements. How was your counting ?
Imagine trying to figure out the number of arrangements for 24 books by the same way it was done for three books! The good news is that they have discovered the FCP that gives the same answer: 3*2*1=6. This kind of multiplication by descending order is maybe familiar to you. This is the same as three factorial by this notification, 3!. It is just the way to fundamentally count things and that should suffice to illustrate the point.
You know what? Let’s do some more exercises in counting
There were 12 students in a race and medal were awarded to those who made it in the first, second and third places. How many different ways can 12 runners finish first, second and third places without allowing any ties?
12*11*10 = 1,320. There is a formula that we can use for kind of question that yields the same answer. This is the permutation formula
The order in which these runners finish matters. Who gets the gold medal, for example, is major.
But if the race was for the purpose of selecting the first three runners to advance to the championship race then the order in which the first three runners cross the line does not matter.
COMBINATIONS
A selection in which order does not matter is termed combinations
Need formula
How does permutation differ from combination?
For the former, think about the different ways of arranging thing where order matters.
Whilst for the latter, think about the different ways of choosing things where order does not matter.
In this section we shall deal with real-life math applications along with some brain teasers.
FUNDAMENTAL COUNTING PRINCIPLE (FCP):
So you say you can count! Of course you may can count naturally, without even thinking about it! But can you count probabilistically, without giving it a thought?
You purchased 4 pants and 3 shirts. How many possible ways can you dress?
4*3 ways= 12 possible outfits.
In the formal definition of the (FCP) we call these items, events. Thus M occurs in m ways followed by event N that occurs n ways then event M followed by N can occur in m*n ways.
PERMUTATION is an arrangement of items in which order matters. This concept can be very elusive and so let us make a very basic and practical example.
There are three books to be shelved. Let’s call them book A,B and C for simplicity. How many possible ways you can arrange them on the shelf?
Let us count together this way:
1. ABC
2. ACB
3. BCA
4. BAC
5. CBA
6. CAB
Clearly, what books you choose to put first, second or third make a difference because order matters. These are really six distinct ways that you could have chosen to arrange them on the shelf. So there are six possible choices of arrangements. How was your counting ?
Imagine trying to figure out the number of arrangements for 24 books by the same way it was done for three books! The good news is that they have discovered the FCP that gives the same answer: 3*2*1=6. This kind of multiplication by descending order is maybe familiar to you. This is the same as three factorial by this notification, 3!. It is just the way to fundamentally count things and that should suffice to illustrate the point.
You know what? Let’s do some more exercises in counting
There were 12 students in a race and medal were awarded to those who made it in the first, second and third places. How many different ways can 12 runners finish first, second and third places without allowing any ties?
12*11*10 = 1,320. There is a formula that we can use for kind of question that yields the same answer. This is the permutation formula
The order in which these runners finish matters. Who gets the gold medal, for example, is major.
But if the race was for the purpose of selecting the first three runners to advance to the championship race then the order in which the first three runners cross the line does not matter.
COMBINATIONS
A selection in which order does not matter is termed combinations
Need formula
How does permutation differ from combination?
For the former, think about the different ways of arranging thing where order matters.
Whilst for the latter, think about the different ways of choosing things where order does not matter.
In this section we shall deal with real-life math applications along with some brain teasers.
What exercise is the the physical muscles mathematics is to the mental muscles ( exercising the mind.) Challenge your mind, it would be very happy you did, your mind will began to expand in perceptions and acuity. The change maybe imperceptible, just like you do not see the physical muscle tissues expanding under weight. The result comes over time. Let us start to lift these mental weights in one of the most interesting quantitative fields, mathematics. The result could make you very shrewd and sagacious if you are not already!
Mathematics at the introductory level as taught in most western institutions seems to be a series of mere computations along with some rules and formula to be followed.
The Chicken - Leg Problem
1. A chicken farmer also has some sheep for a total of 27 animals, and the animals have 72 legs in all. How many chickens does the farmer have?
Here is an application of system of equations. Use it to figure this out:
c-for chiken and s-for sheep
clue c+s =27 total animals. Find the other equation and solve the system
2. 15 times 2 = 30, but what is 30 divided by one half (1/2) ?
Introduction to axioms , then questions later
axioms
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