1. Permutations with Repetition

These are the easiest to calculate.
When we have n things to choose from ... we have n choices each time!
When choosing r of them, the permutations are:

n × n × ... (r times)

(In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.)
Which is easier to write down using an exponent of r:

n × n × ... (r times) = n^r

Example: in the lock above, there are 10 numbers to choose from (0,1,...9) and we choose 3 of them:

10 × 10 × ... (3 times) = 10³ = 1,000 permutations

So, the formula is simply:

nr

where n is the number of things to choose from, and we choose r of them (Repetition allowed, order matters)

2. Permutations without Repetition

In this case, we have to reduce the number of available choices each time.

For example, what order could 16 pool balls be in?
After choosing, say, number "14" we can't choose it again.
So, our first choice has 16 possibilites, and our next choice has 15 possibilities, then 14, 13, etc. And the total permutations are:

16 × 15 × 14 × 13 × ... = 20,922,789,888,000

But maybe we don't want to choose them all, just 3 of them, so that is only:

16 × 15 × 14 = 3,360

In other words, there are 3,360 different ways that 3 pool balls could be arranged out of 16 balls.

Without repetition our choices get reduced each time.

But how do we write that mathematically? Answer: we use the "factorial function"

	The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples: 4! = 4 × 3 × 2 × 1 = 24 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 1! = 1
Note: it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together gets us 1, but it helps simplify a lot of equations.

So, when we want to select all of the billiard balls the permutations are:

16! = 20,922,789,888,000

But when we want to select just 3 we don't want to multiply after 14. How do we do that? There is a neat trick ... we divide by 13! ...

16 × 15 × 14 × 13 × 12 ...		= 16 × 15 × 14 = 3,360
13 × 12 ...

Do you see? 16! / 13! = 16 × 15 × 14
The formula is written:

where n is the number of things to choose from, and we choose r of them (No repetition, order matters)

Examples:

Our "order of 3 out of 16 pool balls example" is:

16!	=	16!	=	20,922,789,888,000	= 3,360
(16-3)!		13!		6,227,020,800

(which is just the same as: 16 × 15 × 14 = 3,360)

How many ways can first and second place be awarded to 10 people?

10!	=	10!	=	3,628,800	= 90
(10-2)!		8!		40,320

(which is just the same as: 10 × 9 = 90)

Notation

Instead of writing the whole formula, people use different notations such as these:

Example: P(10,2) = 90

Combinations

There are also two types of combinations (remember the order does not matter now):

1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)
2. No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combinations with Repetition

Actually, these are the hardest to explain, so we will come back to this later.

2. Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!
The easiest way to explain it is to:

• assume that the order does matter (ie permutations),
• then alter it so the order does not matter.

Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order.
We already know that 3 out of 16 gave us 3,360 permutations.
But many of those are the same to us now, because we don't care what order!
For example, let us say balls 1, 2 and 3 are chosen. These are the possibilites:

Order does matter	Order doesn't matter
1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1	1 2 3

So, the permutations will have 6 times as many possibilites.
In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:

3! = 3 × 2 × 1 = 6

(Another example: 4 things can be placed in 4! = 4 × 3 × 2 × 1 = 24 different ways, try it for yourself!)
So we adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren't interested in their order any more):

That formula is so important it is often just written in big parentheses like this:

where n is the number of things to choose from, and we choose r of them (No repetition, order doesn't matter

Mathematical investigation and Modeling reflection

Lengel F. Gonito

BSED III-A1 Mathematics

Mathematical Investigation and Modeling

Mathematical investigation

            A mathematical investigation is typically defined as an investigation that involves a collection of mathematical and problem solving based issues. Such issues generally have multiple uses and purposes. They are typically open-ended in their content. There are generally more than one possible solution, and use more than one method in solving the issue at hand. Mathematical investigations may also consist of statistical data. Mathematical investigations are furthermore centered on some sort of theme or central issue/hypothesis (Marie, 2001).Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended. Mathematical investigation is more of a divergent activity. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation. Ronda (2010) said that the exploration of the situation, the formulation of problems and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalizations.

Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended. Problem solving is a convergent activity. It has definite goal, the solution of the problem. Mathematical investigation on the other hand is more of a divergent activity. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation. The exploration of the situation, the formulation of problems and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalizations. It is these thinking processes which enable an individual to learn more mathematics, apply mathematics in other discipline and in everyday situation and to solve mathematical (and non-mathematical) problems. Teaching anchored on mathematical investigation allows for students to learn about mathematics, especially the nature of mathematical activity and thinking. It also make them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning, etc and not about memorizing and following existing procedures. The ultimate aim of mathematical investigation is develop students’ mathematical habits of mind. Although students may do the same mathematical investigation it is not expected that all of them will consider the same problem from a particular starting point.  The “open-endedness” of many investigations also means that students may not completely cover the entire situation. However, at least for a student’s own satisfaction, the achievement of some specific results for an investigation is desirable. What is essential is that the students will experience the following mathematical processes which are the emphasis of mathematical investigation:

• systematic exploration of the given situation
• formulating problems and conjectures
• attempting to provide mathematical justifications for the conjectures.

Mathematical Modeling   

            Mathematical modeling is generally understood as the process of applying mathematics to a real world problem with a view of understanding the latter. One can argue that mathematical modeling is the same as applying mathematics where we also start with a real world problem, we apply the necessary mathematics, but after having found the solution we no longer think about the initial problem except perhaps to check if our answer makes sense. This is not the case with mathematical modeling where the use of mathematics is more for understanding the real world problem. The modeling process may or may not result to solving the problem entirely but it will shed light to the situation under investigation. The figure below shows key steps in modeling process. Mathematical modeling approaches can be categorized into four broad approaches: Empirical models, simulation models, deterministic models, and stochastic models. The first three models can very much is integrated in teaching high school mathematics. The last will need a little stretching.

Empirical modeling involves examining data related to the problem with a view of formulating or constructing a mathematical relationship between the variables in the problem using the available data.

Simulation modeling involves the use of a computer program or some technological tool to generate a scenario based on a set of rules. These rules arise from an interpretation of how a certain process is supposed to evolve or progress.

Deterministic modeling in general involves the use of equation or set of equations to model or predicts the outcome of an event or the value of a quantity.

Stochastic modeling takes deterministic modeling one further step. In stochastic models, randomness and probabilities of events happening are taken into account when the equations are formulated. The reason behind this is the fact that events take place with some probability rather than with certainty. This kind of modeling is very popular in business and marketing.

My Reflection

Mathematical investigation and modeling it involves a collection of mathematical and problem solving based issues and the process of applying mathematics to a real world problem with a view of understanding the latter.

Mathematical investigation and modeling is one of those subjects I have taken that presents challenges or the most challenging not only for me but also for my classmates. This subject is a kind of subject that is not as simple as what we are thinking of. When I enrolled at first semester of my 3^rd year level in college one of the subjects that I saw as I fill up the enrollment form is MIM which stands for Mathematical Investigation and Modeling as a part of my chosen major which is mathematics. It really caught my attention. I think for the possibilities or ideas of what this subject is all about. By simply reading the word investigation and modeling, the word immediately come in into my mind is that maybe this subject is difficult and challenging. But instead of thinking negative things I used to prepare myself to face the goodness, difficulties or challenges at the back of this unfamiliar major subject.

The most awaited moment finally come, the day when I first encounter Mathematical Investigation and Modeling. Our professor defines it as a process of collection of mathematical and problem solving based issues and the process of applying mathematics to a real world problem with a view of understanding the important concepts and problems. Through further discussion I learned lots of things about it, yes indeed MIM is really challenging. Critical thinking, analytic thinking and deeper understanding are the requirements for you to engage yourself to this field.

Time goes by as we go through studying MIM I have encountered challenges, I started to experience solving problems which have higher level of difficulty compare to the common mathematical problem, those problems under this subject really arouse our mental abilities. But the good thing is that I am no alone as I faced those challenges I or in solving those mathematical problems, our professor allows us to have our groups to solve problems together and with the help of every group member. For this instance although the complexity is still there the tasks within solving is now more exciting because you are working with you group, the difficulty of conducting investigation in solving such situation is lessen because there are multiple numbers of ideas coming from different minds that united or converged. Little by little I have started to cope with the lesson and with this subject.  Many weeks and months had passed I started to enjoyed taking this subject, maybe because through this I have meet my progress in engaging myself to the new level of solving problems. Another reason is that through this I started to adopted new environment of studying mathematical problems which improved my mental ability though I can’t avoid feeling confused and headache brought by the tricky word problems. Another good thing I realized after taking this subject is that it molded great cooperation and collaboration within everyone of us, as we work together. It made us to realize the importance of sharing ideas then mixing them together to come up to the better result.

Being part of this is what we called modeling, which refers to application of the mathematical problems or constructing word problems with its relation to the real life set up or application. In here I learned how to connect mathematical problem to the situation wherein I or we commonly observed in our daily lives. Through this I have noticed and proven that mathematics is really relevant when talking about its connection to the life of people. This subject is a factual proof about the importance of mathematics in the life of every one of us, that its existence is not just to have this for us to study about but it has an important function not only in education but also in real life situation.

The bloody experience in solving word problems finally ended up, when midterm part is finally over as well. It was nice to think that we have passed that level of difficulty and hardship but breaking news suddenly heard by my ears. Our adviser told us that we have to conduct a demonstration teaching as a part of our requirements on his subject MIM, and for us to apply the knowledge and learning we have acquired. We are asked to select topics we are going to used and construct an investigation and modeling regarding that topic. It is hard to make that one because we are required to make investigation and modeling which is new and unique to our professor.  We are all challenge to make this for the sake of our grades. We find difficulty in this kind of task, what we have conducted first is making lesson plan which was MIM approach lesson plan and then have its application by means of demonstration. Demonstration began, nervous among my classmates can be observed until the day of my turn finally arrived. It served as a big challenge to my part because many of my classmates who are done in their demonstration got a good and best result. The rats inside my chest started to run, I don’t even know how this presentation happened but all I did is just built the trust to me and potentials. This is really worth it, I thanked God because I got a good result though my grade is not as high as what others got, I feel contented and deserved to what I work for. This experience taught me more things. It served as my good practice on how I will handle my future, especially when I reach the moment of being a successful teacher.

To sum it up, I therefore conclude that having this subject as part of my chosen course really helpful. It enhanced one’s ability especially when it comes to mathematics. It serves as an training and practice especially to those who are in the pasture of studying mathematics as their field of specialization. Although the suffering is there but it’s just a tool or part of challenges that awakened one’s mind to achieved their progress regarding their capability, as a mathematician and great problem solver. It made me to realize that learning enables us to do things we were never able to do, change our perception of the world and our relationship to it, and extend our capacity to create.