Permutation, Combination & Probability¶
Factorial Notation¶
Let \(n\) be a positive integer. Then the product \(n\)\((n-1)\)\((n-2)\)....\(3.2.1\) is called it factorial.
Symbol¶
In above case \(n\) factorial is denoted by \(n!\) and it can also be written as \(\underline{|x}\)
Examples¶
\(0!\) =\(1\) (by definition)
\(1!\) =\(1\)
\(2!\) =\(2.1\) = \(2\)
\(3!\) =\(3.2.1\) = \(6\)
\(4!\) =\(4.3.2.1\) = \(24\)
\(5!\) =\(5.4.3.2.1\) = \(120\)
\(6!\) =\(6.5.4.3.21\) = \(720\)
\(7!\) =\(5040\)
\(8!\) =\(40320\)
\(9!\) =\(362880\)
\(10!\) =\(3628800\)
Note
- \((a+b)!\rlap{/}{=}a!+b!\)
- \((a\cdotp b)!\rlap{/}{=}a!\cdotp b!\)
Principles of Counting¶
Fundamental Principle of Addition¶
If an operation can be performed in \(m\) ways and another operation can be performed in \(n\) ways, independent of first operation then either of the two operations can be performed in \(m+n\) ways. This is called fundamental principal of addition.
Fundamental Principle of Multiplication¶
If an operation can be performed in \(m\) ways and afterwards a second Operation can be performed in \(n\) ways then both these operations can be performed in \(m \times n\) ways. This is called fundamental principle of multiplication.
Note
These principles can be extended tip to finite number of operations.
Permutations¶
The different arrangements which can be made out of a given number. taken one or more at a time. are called permutations.
OR
An arrangement of a finite number of objects one or more at a time is called a permutation of these objects (w.r.t. their order).
Rules¶
- The number of Permutations of n-distinct things taken all at a time is \(n!\).
- The number of permutations of \(n\) distinct objects taken \(r\) at a time is given by: $$ ^nP_r=\frac{n!}{(n-r)!}\;\; 0\le r\le n $$
- \(^nP_r=n!\)
- \(^nP_0=1\)
- If one of \(n\) given objects \(p\) are alike and of one, kind, \(q\) are alike and of second kind, \(r\) are alike and of third kind, and so on then the number of permutations of these n-objects taken all at a time is given by: $$ \dbinom{n!}{p!q!r!} $$ formula can also be written as: $$ \dbinom{n}{pqr} $$
- The number of circular permutation of \(n\) different objects taken all at a time is \((n-1)!\)
- Number of arrangements of n=beads (keys) etc, can be arranged on circular wire (key ring) is $$ \frac{(n-1)!}{2} $$
Combinations¶
The different groups or selections which can be made out of a given number of objects, taken one or more at a time (irrespective of their arrangements) are called combinations.
Rules¶
- The number of combinations of \(n\) distinct objects, taken all at a time is 1.
- The number of combinations of \(n\) distinct objects taken \(r\) at a time is given by: $$ ^nC_r=\dbinom{n}{r}=\frac{n!}{r!(n-r)!} $$.
- The number of combinations of \(n\) distinct objects taken \(r\) at a time when \(p\) particular objects are always included is given by: $$ ^{n-p}C_{r-p} $$
- The number of combinations of \(n\) distinct objects taken \(r\) at a time when \(p\) particular objects are never included is given by: $$ ^{n-p}C_r $$
- \(^nC_0=^nC_n=1\)
- \(^nC_r=^nC_{r-1}\)
- \(^nC_r+^nC_{r-1}=^{n+1}C_r\)
- If \(^nC_r=^nC_s\), then either \(r=s\) or \(r+s=n\)
- \(\frac{^nC_r}{^nC_{r-1}}=\frac{n-r+1}{r}\)
- \(^nC_r.r!=^nP_r\)
- \(^nC_0+^nC_1+nC_2+\)...\(+^nC_n=2^n\)
- \(^nC_0+^nC_2+nC_4+\)...=\(^nC_1+^nC_3+nC_5+\)...\(=2^{n-1}\)
- The number of ways of selecting one or more objects out of \(n\) objects is \(2^n-1\)
Note
In most cases, given problem cannot be determined whether it belongs to permutation or combination. On the basis of few words or on the assumption in expression of language given in the problem. we can identify that problem according to the following table:
| Permutation | Combination |
|---|---|
| Arrangements, Standing or sitting in roe or in a circle. Problem regarding digits, letter, formation of words, numbers etc |
Selection, choice, draw etc. Distribution, formation of a group, committee, team etc. Problem regarding geometry. |
Probability¶
Probability is the numerical evaluation of a chance that a particular event would occur.
Definitions¶
Random experiment¶
An experiment in which all possible outcomes are known in advance.
Sample space¶
The set of all possible outcomes of an experiment is called the sample space of the experiment. It is denoted by \(S\)
Event¶
A subset of the sample space is called an event.
Mutually exclusive or disjoint events¶
Two events \(A\) & \(B\) are said to be mutually exclusive or s=disjoint if and only if they cannot both occur at same time i.e they have no point in common $$ A\cap B=\phi $$
Equally likely events¶
Two events \(A\) and \(B\) are said to be equally likely when one event is as likely to occur as the other.
Dependent / Independent events¶
Two or more events such that occurrence of any one does not depends on the occurrence of any other, they are said to be independent events. Other wise dependent events.
Results¶
- The probability of happening of an event \(E\) is defined as: $$ P(E)=\frac{n(E)}{n(S)} $$
- Probability of not happening of an event: $$ P(\overline{E})=1-P(E) $$
- \(0\le P(E)\le 1\)
- \(P(E)+P(\overline{E})=1\)
- If \(E\) is an impossible event then \(P(E)=0\)
- If \(E\) is certain event then \(P(E)=1\)
- Addition theorem: for any two event \(A\) & \(B\) $$ P(A\cup B)=P(A)+P(B)- P(A\cap B) $$
- If \(A\) & \(B\) are mutually exclusive events then $$ P(A\cup B)=P(A)+P(B) $$
- Multiplication law: if \(A\) & \(B\) are two independent events, then the probability that both the events occur simultaneously in any order is given as: $$ P(A\cap B)=P(A)\times P(B) $$