Permutation and Combination

Permutations and Combinations-

‘Permutations and Combinations’ is the next post of my series Topics in IB Mathematics.It is very useful and interesting as a topic. It’s also very useful in solving problems of Probability. To understand Permutations and Combinations, we first need to understand Factorial.

Definition of Factorial-  If we multiply n consecutive natural numbers together, then the product is called factorial of n. Its shown by n! or by

for example :       n! = n(n - 1)(n - 2)(n - 3)..........3.2.1

Some Properties of Factorials-
(i) Factorials can only be calculated for positive integers at this level. We use gamma functions to define non-integer factorial that’s not required at this level
(ii) Factorial of a number can be written as a product of that number with the factorial of its predecessor    n! = n[(n - 1)(n - 2)(n - 3)..........3.2.1]

 = n(n - 1)!

(iii)  0! = 1  you can watch this video for the explanation.

(iv)  If we want to simplify a “permutations and combinations” expression that has factorials in the numerator as well as in the denominator, we make all the factorials equal to the smallest factorial

Exponent of Prime Number p in n!- Let’s assume that p is a prime number and n is a positive integer, then exponent of p in n! is denoted by  Ep (n!)

{E_p}(n!) = \left[ {\frac{n}{p}} \right] + \left[ {\frac{n}{{{p^2}}}} \right] + ........\left[ {\frac{n}{{{p^t}}}} \right]

We can’t use this result to find the exponent of composite numbers.

Fundamental Principle of Counting- In almost all IB Mathematics books, the first exercise of Permutations and Combinations is based on Fundamental Principle of Counting. We can learn it in two steps.

Principle of Addition- If there are x different ways to do a work and y different ways to two another work and both the works are independent of each other then there are (x+y) ways to do either first OR second work

Example- If we can choose a man in a team by 6 different ways and a woman by 4 different ways then we can choose either a man or a woman by 6+4=10 different ways.

Principle of Multiplication-If there are x different ways to do a work and y different ways to two another work and both the works are independent of each other then there are (x.y) ways to do both first AND second works.

Example- If we can choose a man in a team by 6 different ways and a woman by 4 different ways then we can choose a man and a woman by 6*4=24 different ways.

Definition of Permutation- Process of making different arrangements of objects, letters and words etc by changing their position is known as permutation

Example- A, B, and C are four books then we can arrange them BY 6 DIFFERENT WAYS  ABC, ACB BCA, BAC CAB CBA. so we can say that there are 6 different permutations of this arrangement.

Number of Permutations of n different objects taken all at a time- If we want to arrange n objects at n different places then the total number of ways of doing this or the total number of permutations = {}^n{p_n}

=n! here P represents permutations

Number of Permutations of n different objects taken r at a time- If we want to arrange n objects at r different places then the total number of ways of doing this or the total number of permutations =  {}^n{p_r}

=  \frac{{n!}}{{\left( {n - r} \right)!}}    here  {}^n{p_r} represent permutations of n objects taken r at a time.

Number of Permutations of n  objects when all objects are not different- If we have n objects in total out of which p are of one type, q are of another type, r are of any other type, remaining objects are all different from each other, the total number of ways of arranging them=  \frac{{n!}}{{p!q!r!}}

Number of Permutations of n different objects taken all at a time when repetition of objects is allowed-If we want to arrange n objects at n different places and we are free to repeat objects as many times as we wish, then the total number of ways of doing this or the total number of permutations = {n^n}

Number of Permutations of n different objects taken r at a time when repetition of objects is allowed- If we want to arrange n objects at r different places(taking r at a time) and we are free to repeat objects as many times as we wish, then the total number of ways of doing this or the total number of permutations= {n^r}

Circular Permutations- When we talk about arrangements of objects, it usually means linear arrangements. But if we wish, we can also arrange objects in a loop. Like we can ask our guests to sit around a round dining table. These types of arrangements are called circular permutations.

If we want to arrange n objects in a circle, then the total number of ways/circular permutations=(n-1)! this case works when there is some difference between clock-wise and anti-clockwise orders

IF there is no distinction between clock-wise and anti-clockwise orders, the total number of permutations=(n-1)!/2

Restricted Permutations- there may be following cases of restricted permutation

(a)   Number of arrangements of ‘n’ objects, taken ‘r’ at a time, when a particular object is to be always included     = r{.^{n - 1}}{P_{r - 1}}

(b) Number of arrangements of ‘n’ objects, taken ‘r’ at a time, when a particular object is fixed: =  ^{n - 1}{P_{r - 1}}

(c) The number of arrangements of ‘n’ objects, taken ‘r’ at a time, when a particular object is never taken: = n-1 Pr.

(d) The number of arrangements of ‘n’ objects, taken ‘r’ at a time, when ‘m’ specific objects always come with each-other =  m!{\rm{ }}{\rm{.}}\left( {{\rm{ }}n - m + 1} \right){\rm{ }}!

(e) The number of arrangements of ‘n’ things, taken all at a time, when ‘m’ specific objects always come with each other= \begin{array}{lllllllllllllll} {n{\rm{ }}!{\rm{ }} - {\rm{ }}\left[ {{\rm{ }}m!\left( {n - m + 1} \right)!{\rm{ }}} \right]}\\ \end{array}

In my next post, I will discuss in detail about combinations and will share a large worksheet based on  P & C. In the meantime you can download and solve these questions

 Permutation Worksheet.pdf

 

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