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Each of the different groups or selections which can be formed by taking some or all of a number of objects, irrespective of their arrangements, is called a combination.   Notation : The number of all combinations of \[n\] things, taken \[r\] at a time is denoted by \[C\,(n,\,r)\] or \[^{n}{{C}_{r}}\,\text{or }\left( \begin{matrix} n  \\ r  \\ \end{matrix} \right)\]. \[^{n}{{C}_{r}}\] is always a natural number.   Difference between a permutation and combination :   (i) In a combination only selection is made whereas in a permutation not only a selection is made but also an arrangement in a definite order is considered.   (ii) Each combination corresponds to many permutations. For example, the six permutations \[ABC,\text{ }ACB,\text{ }BCA,\text{ }BAC,\text{ }CBA\] and \[CAB\] correspond to the same combination \[ABC\].

In circular permutations, what really matters is the position of an object relative to the others.   Thus, in circular permutations, we fix the position of the one of the objects and then arrange the other objects in all possible ways.   There are two types of circular permutations :   (i) The circular permutations in which clockwise and the anticlockwise arrangements give rise to different permutations, e.g. Seating arrangements of persons round a table.   (ii) The circular permutations in which clockwise and the anticlockwise arrangements give rise to same permutations, e.g. arranging some beads to form a necklace.   Difference between clockwise and anti-clockwise arrangement : If anti-clockwise and clockwise order of arrangement are not distinct e.g., arrangement of beads in a necklace, arrangement of flowers in garland etc. then the number of circular permutations of n distinct items is \[\frac{(n-1)\,!}{2}\].   (i) Number of circular permutations of n different things, taken  \[r\] at a time, when clockwise and anticlockwise orders are taken as different is\[\frac{^{n}{{P}_{r}}}{r}\].   (ii) Number of circular permutations of \[n\] different things, taken \[r\] at a time, when clockwise and anticlockwise orders are not different is \[\frac{^{n}{{P}_{r}}}{2r}\].   Theorems on circular permutations   Theorem (i) : The number of circular permutations of n different objects is \[(n-1)\,!\].   Theorem (ii) : The number of ways in which n persons can be seated round a table is \[(n-1)!\].   Theorem (iii) : The number of ways in which n different beads can be arranged to form a necklace, is \[\frac{1}{2}(n-1)!\].

(1) Number of permutations of \[n\] dissimilar things taken \[r\] at a time when p particular things always occur \[=\,{{\,}^{n-p}}{{C}_{r-p}}r!\].   (2) Number of permutations of \[n\] dissimilar things taken \[r\] at a time when \[p\] particular things never occur \[{{=}^{n-p}}{{C}_{r}}\,r!\].   (3) The total number of permutations of \[n\] different things taken not more than \[r\] at a time, when each thing may be repeated any number of times, is \[\frac{n({{n}^{r}}-1)}{n-1}\].   (4) Number of permutations of \[n\] different things, taken all at a time, when \[m\] specified things always come together is \[m\,!\,\times \,\,(n-m+1)\,!\].   (5) Number of permutations of \[n\] different things, taken all at a time, when \[m\] specified things never come together is \[n\,!-m\,!\,\,\times \,\,(n-m+1)\,!\].   (6) Let there be \[n\] objects, of which \[m\] objects are alike of one kind, and the remaining \[(n-m)\] objects are alike of another kind. Then, the total number of mutually distinguishable permutations that can be formed from these objects is \[\frac{n!}{(m\,!)\,\,\times \,\,(n-m)\,!}\].   The above theorem can be extended further i.e., if there are \[n\] objects, of which \[{{p}_{1}}\] are alike of one kind; \[{{p}_{2}}\] are alike of another kind; \[{{p}_{3}}\] are alike of \[{{3}^{rd}}\] kind;......; \[{{p}_{r}}\] are alike of \[{{r}^{th}}\] kind such that \[{{p}_{1}}+{{p}_{2}}+......+{{p}_{r}}=n\]; then the number of permutations of these n objects is \[\frac{n\,!}{({{p}_{1}}\,!)\times ({{p}_{2}}\,!)\times ......\times ({{p}_{r}}!)}\].  

(1) The number of permutations (arrangements) of \[n\] different objects, taken \[r\] at a time, when each object may occur once, twice, thrice,........upto r times in any arrangement = The number of ways of filling \[r\] places where each place can be filled by any one of \[n\] objects.           The number of permutations = The number of ways of filling \[r\] places \[={{(n)}^{r}}\].   (2) The number of arrangements that can be formed using \[n\] objects out of which \[p\] are identical (and of one kind) \[q\] are identical (and of another kind), \[r\] are identical (and of another kind) and the rest are distinct is \[\frac{n\,!}{p!\,q!r!}\].

(1) Arranging \[n\] objects, taken \[r\] at a time equivalent to filling \[r\] places from \[n\] things.         The number of ways of arranging = The number of ways of filling \[r\] places.   \[=n(n-1)\,(n-2).......(n-r+1)\]     \[=\frac{n(n-1)\,(n-2).....(n-r+1)((n-r)!)}{(n-r)!}=\frac{n\,!}{(n-r)!}{{=}^{n}}{{P}_{r}}\]   (2) The number of arrangements of \[n\] different objects taken all at a time \[={{\,}^{n}}{{P}_{n}}=n!\]   (i) \[^{n}{{P}_{0}}=\frac{n\,!}{n\,!}=1;{{\,}^{n}}{{P}_{r}}=n\,{{.}^{n-1}}{{P}_{r-1}}\]   (ii) \[0\,!=1;\,\,\frac{1}{(-r)\,!}=0\] or \[(-r)\,!=\infty \,\,\,(r\in N)\]

The ways of arranging or selecting a smaller or an equal number of persons or objects at a time from a given group of persons or objects with due regard being paid to the order of arrangement or selection are called the (different) permutations.   For example : Three different things \[a,\,\,b\] and \[c\] are given, then different arrangements which can be made by taking two things from three given things are \[ab,ac,\text{ }bc,\text{ }ba,\text{ }ca,\text{ }cb\].   Therefore the number of permutations will be 6.

(1) The Factorial : Factorial notation: Let \[n\] be a positive integer. Then, the continued product of first \[n\] natural numbers is called factorial n, to be denoted by \[n\,!\] or \[\left| \!{ {\,n \,}} \right. \].   Also, we define \[0!=1\].   when \[n\] is negative or a fraction, \[n\,\,!\] is not defined.   Thus, \[n\,\,!=n(n-1)\,(n-2)\,....\,3.2.1.\]   (2) Exponent of Prime \[p\] in \[n\,!\] : Let \[p\] be a prime number and \[n\] be a positive integer. Then the last integer amongst 1, 2, 3, ....... \[(n-1),\,\,n\] which is divisible by \[p\] is \[\left[ \frac{n}{p} \right]p\], where \[\left[ \frac{n}{p} \right]\] denotes the greatest integer less than or equal to \[\frac{n}{p}\].

Algebraic expression containing many terms of the form \[c{{x}^{n}},\,\,n\] being a non-negative integer is called a polynomial. i.e., \[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+{{a}_{3}}{{x}^{3}}+......+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n}}{{x}^{n}}\], where \[x\] is a variable, \[a>0,\,(<0)\] are constants and \[{{a}_{n}}\ne 0\].    Example : \[4{{x}^{4}}+3{{x}^{3}}-7{{x}^{2}}+5x+3\], \[3{{x}^{3}}+{{x}^{2}}-3x+5\].   (1) Real polynomial   \[f(x)={{a}_{0}}+{{a}_{1}}x+\]\[{{a}_{2}}{{x}^{2}}+\]\[{{a}_{3}}{{x}^{3}}+\]\[.....+{{a}_{n}}{{x}^{n}}\]   is called real polynomial of real variable x with real coefficients.   Example: \[3{{x}^{3}}-4{{x}^{2}}+5x-4,\,{{x}^{2}}-2x+1\]   etc. are real polynomials.   (2) Complex polynomial   \[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+{{a}_{3}}{{x}^{3}}+\]\[......+{{a}_{n}}{{x}^{n}}\]   is called complex polynomial of complex variable \[x\] with complex coefficients.   Example: \[3{{x}^{2}}-(2+4i)x+(5i-4),\,{{x}^{3}}-5i\,{{x}^{2}}+(1+2i)x+4\] etc. are complex polynomials.   (3) Degree of polynomial : Highest power of variable x in a polynomial is called degree of polynomial.   Example: \[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+......+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n}}{{x}^{n}}\] is a \[n\] degree polynomial.   \[f(x)=4{{x}^{3}}+3{{x}^{2}}-7x+5\] is a 3 degree polynomial.   A polynomial of second degree is generally called a quadratic polynomial. Polynomials of degree 3 and 4 are known as cubic and biquadratic polynomials respectively.   (4) Polynomial equation : If \[f(x)\] is a polynomial, real or complex, then \[f(x)=0\] is called a polynomial equation.  

An equation in which the highest power of the unknown quantity is two is called quadratic equation.   Quadratic equations are of two types :  

Purely quadratic	Adfected quadratic
\[a{{x}^{2}}+c=0\], where \[a,c\in C\] and \[b=0,\,a\ne 0\]	\[a{{x}^{2}}+bx+c=0\],  where \[a,\,b,\,c\in C\] and \[a\ne 0,\,\,b\ne 0\]

  Roots of a quadratic equation : The values of variable \[x\] which satisfy the quadratic equation is called roots of quadratic equation.  

(1) Factorization method   Let \[a{{x}^{2}}+bx+c=\]\[a(x-\alpha )(x-\beta )=0\].   Then \[x=\alpha \] and \[x=\beta \] will satisfy the given equation.   Hence, factorize the equation and equating each factor to zero gives roots of the equation.   Example : \[3{{x}^{2}}-2x+1=0\] \[\Rightarrow \]\[(x-1)(3x+1)=0\];   \[x=1,\,-1/3\]   (2) Sri Dharacharya method : By completing the perfect square as  \[a{{x}^{2}}+bx+c=0\]\[\Rightarrow \]\[{{x}^{2}}+\frac{b}{a}x+\frac{c}{a}=0\]   Adding and subtracting \[{{\left( \frac{b}{2a} \right)}^{2}}\], \[\left[ {{\left( x+\frac{b}{2a} \right)}^{2}}-\frac{{{b}^{2}}-4ac}{4{{a}^{2}}} \right]=0\]   which gives, \[x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].   Hence the quadratic equation \[a{{x}^{2}}+bx+c=0\] \[(a\ne 0)\] has two roots, given by \[\alpha =\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}\], \[\beta =\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}\]   Every quadratic equation has two and only two roots.