Permutations and Combinations: Lecture 1

In this lecture, we learn about the following concepts.

1. the addition and multiplication principle
2. permutations

• permutation with distinct objects
• permutation with identical objects
• permutation allowing repetition
• circular permutation

The addition and multiplication principle

• The addition principle is used in the case that you have a few different ways to complete a task.
• The multiplication principle is used in the case that you have different stages in a task and you have a few ways to complete each stage.

Permutation with distinct objects

• When you choose r objects from n distinct projects (the order of choosing matters), the no. of ways that you can do so is \[_{}^{n}\textrm{P}_{r}\]

In the another way, we can use the multiplication principle. To choose the first object, we have n choices , i.e. n no. of ways. To choose the second object, we are left with (n-1) choices, i.e. (n -1) ways. Hence, the no. of ways to choose r objects is n(n-1)(n-2)...(n-r+1)

• In the GC, you can find "nPr" in MATH -> PRB

Permutation with identical objects

• If we are to arrange r objects with r1 of type 1, r2 of type 2,... and rn of type n, where r1+r2+...rn=r, the no. of ways is given by r!/(r1!r2!...rn!)

• The rationale behind this formula:

r! is the no. of ways to arrange the objects if all of the objects are different. If some of them are identical, we actually overcount the no. of ways by a factor, which is the no. of ways we can arrange those identical objects.

Permutation allowing repetition

• If we choose r objects taken from n distinct objects in order, and each object can be used more than once, the no. of ways to do so is \[n^{r}\], since each time when we choose an object, we have n choices.

Circular permutation

• The no. of ways to arrange n distinct objects in a circle is (n-1)!
• The rationale behind the formula:

First of all, we can consider the permutation of n distinct objects, which is n!. However, when we arrange them in a circle, we found that some of the arrangement becomes identical. In fact, since we only care about the order in the clockwise manner and ignore the starting position, every identical arrange is repeated n times. Hence, in the end, the no. of different arrangement is n!/n, which is (n-1)!

Remark: circular permutation with identical objects do not have a general formula. You might want to think about the reason.

Differential Equation: Lecture 4

In today's lecture, we learnt more about the application of DE in real-life.

1. Radioactive decay (You need to understand the term "half life", which means the time taken for the material to distintegrate into half of its original amount. )

2. Newton's law of cooling (The rate of decreasing of a object's temperature is directly proportional to the temperature difference between the object and the surrounding. )

3. Change of volume with time.

For this type of questions, you can follow the steps below to solve the problem.

- Look for key phrases that gives you the differential equation, which is usually a statement on the "rate of increase/decrease".

- Form the DE and solve it to have a general solution. (A solution might have one or two unknown constants.)

-Look for other data given in the questions. Use these data to solve for the unknown constants in the previous step.

By now we have finished the chapter DE, you can attempt all the tutorial questions.

Differential Equation: Lecture 3

In lecture 3, you learnt about two main concepts.

1. What is the family of curves?

2. How to model a question using DE and solve the problem?

1. Family of curves

The general solution of a DE can be represented by a set of curves graphically ( one curve for each value of the arbitrary constant). This set of curve is called the family of solution curves of the DE.

A particular solution of the DE corresponds to one specific curve in the family.

When you are asked to sketch the family of curves, you usually choose C=-1, 0 and 1 unless otherwise specified by the question.

Please use G.C. to help you in the sketching!!!

2. Modelling using DE

DE can be used to model a lot of real-life problems including physical phenomenon, process, human behaviours and population growth.

To model a problem with DE, follow the steps below.

Step 1: look for phrases like " the rate of increasing/decreasing of ... is proportional to ..." or ".... is increasing/decreasing at the rate proportional to... "

These phrases give you the Differential Equation.

Step 2: Solve the DE, usually by direct integration. The solution usually has some unknown constants.

Step 3: Use the given data in the question to find out the unknown constants.

After lecture 3, you will be able to do the whole tut11b if you are good at modelling. However you might wanna attempt Q1, 2, 3 , 6 first since they are similar to what we have gone through in the lecture.

Differential Equation: Lecture 2

In lecture 2, we learnt two main skills.

1. How to solve a second order DE in the form of \[\frac{\mathrm{d^2}y }{\mathrm{d} x^2}=f(x)\]

2. How to use substitution to solve a first order DE that cannot be solved by direct integration.


1. How to solve a second order DE

Step 1: integrate both sides once to get a first order DE (the first order DE obtained will have one arbitrary constant.)

Step 2: integrate again to get the general solution. (the general solution will have 2 arbitrary constant.)

2. How to use substitution to solve a first order DE that cannot be solved by direct integration.

Step 1: differentiate the given formula for substitution. (e.g. z=1/y)

Step 2: replace the dy/dx in the original DE by dz/dx

Step 3: observe whether there are still y in the DE, replace the y using substition or other methods.

The final DE should be in term of x and z and should be solvable by direct integration.

Step 4: solve the DE to obtain a relation between x and z.

Step 5: replace z by y using the substition given in the question.

After lecture 2, you can do the whole tutorial 11a.