Current Affairs JEE Main & Advanced

The rule by which any power of binomial can be expanded is called the binomial theorem.   If \[n\] is a positive integer and \[x,\,\,y\,\,\in C\]  then   \[{{(x+y)}^{n}}{{=}^{n}}{{C}_{0}}{{x}^{n-0}}{{y}^{0}}{{+}^{n}}{{C}_{1}}{{x}^{n-1}}{{y}^{1}}+{{\,}^{n}}{{C}_{2}}\,{{x}^{n-2}}{{y}^{2}}+........\]\[{{+}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}+......{{+}^{n}}{{C}_{n-1}}x{{y}^{n-1}}{{+}^{n}}{{C}_{n}}{{x}^{0}}{{y}^{n}}\]   i.e.,   \[{{(x+y)}^{n}}=\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}.{{x}^{n-r}}.{{y}^{r}}}\]                             .....(i)   Here \[^{n}{{C}_{0}},\,{{}^{n}}{{C}_{1}},{{\,}^{n}}{{C}_{2}},{{......}^{n}}{{C}_{n}}\] are called binomial coefficients and \[{{(1+x)}^{n}}=1+nx+\frac{n(n-1)}{2!}{{x}^{2}}+.......\] for \[0\le r\le n\].

An algebraic expression consisting of two terms with \[+ve\] or   \[-ve\] sign between them is called a binomial expression.   For example : \[(a+b),\,(2x-3y),\,\left( \frac{p}{{{x}^{2}}}-\frac{q}{{{x}^{4}}} \right)\,,\,\left( \frac{1}{x}+\frac{4}{{{y}^{3}}} \right)\] etc.

Let \[N=p_{1}^{{{\alpha }_{1}}}.p_{2}^{{{\alpha }_{2}}}.p_{3}^{{{\alpha }_{3}}}......p_{k}^{{{\alpha }_{k}}}\], where \[{{p}_{1}},\,{{p}_{2}},{{p}_{3}},......{{p}_{k}}\] are different primes and \[{{\alpha }_{1}},\,{{\alpha }_{2}},\,{{\alpha }_{3}},......,{{\alpha }_{k}}\] are natural numbers then :   (1) The total number of divisors of N including 1 and N is = \[({{\alpha }_{1}}+1)\,({{\alpha }_{2}}+1)\,({{\alpha }_{3}}+1)....({{\alpha }_{k}}+1)\]   (2) The total number of divisors of N excluding 1 and N is = \[({{\alpha }_{1}}+1)\,({{\alpha }_{2}}+1)\,({{\alpha }_{3}}+1).....({{\alpha }_{k}}+1)-2\].   (3) The total number of divisors of N excluding 1 or N is = \[({{\alpha }_{1}}+1)\,({{\alpha }_{2}}+1)\,({{\alpha }_{3}}+1).....({{\alpha }_{k}}+1)-1\].   (4) The sum of these divisors is       \[=(p_{1}^{0}+p_{1}^{1}+p_{1}^{2}+......+p_{1}^{{{\alpha }_{1}}})\,(p_{2}^{0}+p_{2}^{1}+p_{2}^{2}+...+p_{2}^{{{\alpha }_{2}}}).....\]   \[(p_{k}^{0}+p_{k}^{1}+p_{k}^{2}+....+p_{k}^{{{\alpha }_{k}}})\]   (5) The number of ways in which N can be resolved as a product of two factors is   \[\left\{ \begin{matrix} \frac{1}{2}({{\alpha }_{1}}+1)\,({{\alpha }_{2}}+1)....({{\alpha }_{k}}+1),\,\text{If }N\text{ is not a perfect square}  \\ \frac{\text{1}}{2}[({{\alpha }_{1}}+1)\,({{\alpha }_{2}}+1).....({{\alpha }_{k}}+1)+1],\,\text{If }N\text{ is a perfect square}  \\ \end{matrix} \right.\]   (6) The number of ways in which a composite number \[N\] can be resolved into two factors which are relatively prime (or co-prime) to each other is equal to \[{{2}^{n-1}}\] where \[n\] is the number of different factors in \[N\].

Let \[{{x}_{1}},\,{{x}_{2}},\,.......,{{x}_{m}}\] be integers. Then number of solutions to the equation \[{{x}_{1}}+{{x}_{2}}+......+{{x}_{m}}=n\]                                                                                                         .....(i)   Subject to the condition   \[{{a}_{1}}\le {{x}_{1}}\le {{b}_{1}},\,{{a}_{2}}\le {{x}_{2}}\le {{b}_{2}},.......,{{a}_{m}}\le {{x}_{m}}\le {{b}_{m}}\]          .....(ii)   is equal to the coefficient of \[{{x}^{n}}\]in   \[({{x}^{{{a}_{1}}}}+{{x}^{{{a}_{1}}+1}}+......+{{x}^{{{b}_{1}}}})\,({{x}^{{{a}_{2}}}}+{{x}^{{{a}_{2}}+1}}+.....+{{x}^{{{b}_{2}}}})......({{x}^{{{a}_{m}}}}+{{x}^{{{a}_{m+1}}}}+.....+{{x}^{{{b}_{m}}}})\].....(iii)   This is because the number of ways, in which sum of m integers in (i) equals n, is the same as the number of times \[{{x}^{n}}\] comes in (iii).   (1) Use of solution of linear equation and coefficient of a power in expansions to find the number of ways of distribution : (i) The number of integral solutions of \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+......+{{x}_{r}}=n\] where \[{{x}_{1}}\ge 0,\,{{x}_{2}}\ge 0,\,......{{x}_{r}}\,\ge 0\] is the same as the number of ways to distribute n identical things among r persons.   This is also equal to the coefficient of \[{{x}^{n}}\] in the expansion of \[{{({{x}^{0}}+{{x}^{1}}+{{x}^{2}}+{{x}^{3}}+......)}^{r}}\]   = coefficient of \[{{x}^{n}}\] in \[{{\left( \frac{1}{1-x} \right)}^{r}}\]   = coefficient of \[{{x}^{n}}\] in \[{{(1-x)}^{-r}}\]   = coefficient of \[{{x}^{n}}\] in   \[\left\{ 1+rx+\frac{r(r+1)}{2!}{{x}^{2}}+....+\frac{r(r+1)\,(r+2)....(r+n-1)}{n\,!} \right.{{x}^{n}}+......\]   \[=\frac{r(r+1)\,(r+2)....(r+n-1)}{n\,!}=\frac{(r+n-1)!}{n!(r-1)!}{{=}^{n+r-1}}{{C}_{r-1}}\].   (2) The number of integral solutions of \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+\] \[.....+{{x}_{r}}=n\] where \[{{x}_{1}}\ge 1,\,{{x}_{2}}\ge 1,.......{{x}_{r}}\ge 1\] is same as the number of ways to distribute n identical things among r persons each getting at least 1. This also equal to the coefficient of \[{{x}^{n}}\] in the expansion of \[{{({{x}^{1}}+{{x}^{2}}+{{x}^{3}}+......)}^{r}}\].   = coefficient of \[{{x}^{n}}\] in \[{{\left( \frac{x}{1-x} \right)}^{r}}\]   = coefficient of \[{{x}^{n}}\] in \[{{x}^{r}}{{(1-x)}^{-r}}\]   = coefficient of \[{{x}^{n}}\] in     \[{{x}^{r}}\left\{ 1+rx+\frac{r\,(r+1)}{2\,!}{{x}^{2}}+...+\frac{r\,(r+1)\,(r+2)...(r+n-1)}{n\,!}{{x}^{n}}+.. \right\}\]   = coefficient of \[{{x}^{n-r}}\] in   \[\left\{ 1+rx+\frac{r(r+1)}{2!}{{x}^{2}}+...+\frac{r(r+1)(r+2)....(r+n-1)}{n\,!}{{x}^{n}}+..... \right\}\]   = \[\frac{r(r+1)\,(r+2)......(r+n-r-1)}{(n-r)\,!}\] = \[\frac{r(r+1)\,(r+2).....(n-1)}{(n-r)!}\] = \[\frac{(n-1)\,!}{(n-r)!(r-1)!}{{=}^{n-1}}{{C}_{r-1}}\].

(1) Number of total different straight lines formed by joining the \[n\] points on a plane of which  \[m(<n)\] are collinear is \[^{n}{{C}_{2}}{{-}^{m}}{{C}_{2}}+1\].   (2) Number of total triangles formed by joining the \[n\]  points on a plane of which \[m(<n)\] are collinear is \[^{n}{{C}_{3}}{{-}^{m}}{{C}_{3}}\].   (3) Number of diagonals in a polygon of \[n\] sides is \[^{n}{{C}_{2}}-n\].   (4) If \[m\] parallel lines in a plane are intersected by a family of other \[n\] parallel lines. Then total number of parallelograms so formed is \[^{m}{{C}_{2}}{{\times }^{n}}{{C}_{2}}\,\,i.e.,\frac{mn(m-1)(n-1)}{4}\].   (5) Given \[n\] points on the circumference of a circle, then   (i) Number of straight lines \[{{=}^{n}}{{C}_{2}}\]      (ii) Number of triangles  \[{{=}^{n}}{{C}_{3}}\]   (iii) Number of quadrilaterals \[{{=}^{n}}{{C}_{4}}\].   (6) If n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. Then the number of part into which these lines divide the plane is \[=1+\Sigma n\].   (7) Number of rectangles of any size in a square of \[n\times n\] is \[\sum\limits_{r=1}^{n}{{{r}^{3}}}\] and number of squares of any size is \[\sum\limits_{r=1}^{n}{{{r}^{2}}}\].   (8) In a rectangle of \[n\times p\,\,(n<p)\] number of rectangles of any size is \[\frac{np}{4}(n+1)\,(p+1)\] and number of squares of any size is \[\sum\limits_{r=1}^{n}{(n+1-r)\,(p+1-r)}\].

Any change in the given order of the things is called a derangement.   If \[n\] things form an arrangement in a row, the number of ways in which they can be deranged so that no one of them occupies its original place is \[n\,!\,\left( 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+......+{{(-1)}^{n}}.\frac{1}{n\,!} \right)\].

Case I : (1) The number of ways in which \[n\] different things can be arranged into \[r\] different groups is \[^{n+r-1}{{P}_{n}}\] or n ! \[^{n-1}{{C}_{r-1}}\] according as blank group are or are not admissible.   (2) The number of ways in which  \[n\] different things can be distributed into \[r\] different group is   \[{{r}^{n}}{{-}^{r}}{{C}_{1}}{{(r-1)}^{n}}{{+}^{r}}{{C}_{2}}{{(r-2)}^{n}}-.........+{{(-1)}^{n-1}}{{\,}^{n}}{{C}_{r-1}}\] or Coefficient of \[{{x}^{n}}\] is n ! \[{{({{e}^{x}}-1)}^{r}}\].   Here blank groups are not allowed.   (3) Number of ways in which \[m\times n\] different objects can be distributed equally among \[n\] persons (or numbered groups) = (number of ways of dividing into groups) \[\times \] (number of groups)\[!=\frac{(mn)\,!\,n\,!}{{{(m\,!)}^{n}}n!}=\frac{(mn)\,!}{{{(m!)}^{n}}}\].   Case II : (1) The number of ways in which \[(m+n)\] different things can be divided into two groups which contain \[m\] and \[n\] things respectively is, \[^{m+n}{{C}_{m}}{{.}^{n}}{{C}_{n}}=\frac{(m+n)\,!}{m\,!\,n!},m\ne n\].   Corollary: If \[m=n\], then the groups are equal size. Division of these groups can be given by two types.   Type I : If order of group is not important : The number of ways in which \[2n\] different things can be divided equally into two groups is \[\frac{(2n)!}{2!{{(n!)}^{2}}}\].   Type II : If order of group is important : The number of ways in which 2n  different things can be divided equally into two distinct groups is \[\frac{(2n)!}{2!{{(n!)}^{2}}}\times 2!=\frac{2n!}{{{(n!)}^{2}}}\].   (2) The number of ways in which \[(m+n+p)\] different things can be divided into three groups which contain m, n and p things respectively is   \[^{m+n+p}{{C}_{m}}{{.}^{n+p}}{{C}_{n}}{{.}^{p}}{{C}_{p}}=\frac{(m+n+p)\,!}{m\,!n\,!\,p\,!},\,m\ne n\ne p\].   Corollary : If \[m=n=p\], then the groups are equal size. Division of these groups can be given by two types.   Type I : If order of group is not important : The number of ways in which 3p different things can be divided equally into three groups is \[\frac{(3p)\,!}{3!{{(p!)}^{3}}}\].   Type II : If order of group is important : The number of ways in which \[3p\] different things can be divided equally into three distinct groups is \[\frac{(3p)!}{3!{{(p!)}^{3}}}3!=\frac{(3p)\,!}{{{(p!)}^{3}}}\].   (i) If order of group is not important : The number of ways in which \[mn\] different things can be divided equally into m groups is \[\frac{mn!}{{{(n!)}^{m}}m!}\].   (ii) If order of group is important: The number of ways in which \[mn\] different things can be divided equally into \[m\] distinct groups is \[\frac{(mn)!}{{{(n!)}^{m}}m!}\times m!=\frac{(mn)!}{{{(n!)}^{m}}}\].

(1) The number of ways in which r objects can be selected from n different objects if k particular objects are   (i) Always included = \[^{n-k}{{C}_{r-k}}\]                 (ii) Never included = \[^{n-k}{{C}_{r}}\]   (2) The number of combinations of n objects, of which p are identical, taken r at a time is   \[^{n-p}{{C}_{r}}{{+}^{n-p}}{{C}_{r-1}}{{+}^{n-p}}{{C}_{r-2}}+.......{{+}^{n-p}}{{C}_{0}}\], if \[r\le p\] and   \[^{n-p}{{C}_{r}}{{+}^{n-p}}{{C}_{r-1}}{{+}^{n-p}}{{C}_{r-2}}+.......{{+}^{n-p}}{{C}_{r-p}}\], if \[r>p\].

(1) The number of combinations of \[n\] distinct objects taken \[r\] at a time when any object may be repeated any number of times.   = Coefficient of \[{{x}^{r}}\] in \[{{(1+x+{{x}^{2}}+.......+{{x}^{r}})}^{n}}\]   = Coefficient of \[{{x}^{r}}\] in \[{{(1-x)}^{-n}}{{=}^{n+r-1}}{{C}_{r}}\]   (2) The total number of ways in which it is possible to form groups by taking some or all of \[n\] things at a time is \[^{n}{{C}_{1}}+{{\,}^{n}}{{C}_{2}}+........+{{\,}^{n}}{{C}_{n}}={{2}^{n}}-1\].   (3) The total number of ways in which it is possible to make groups by taking some or all out of \[n=({{n}_{1}}+{{n}_{2}}+....)\] things, when \[{{n}_{1}}\] are alike of one kind, \[{{n}_{2}}\] are alike of second kind, and so on is \[\{({{n}_{1}}+1)\,({{n}_{2}}+1)......\}-1\].   (4) The number of selections of \[r\] objects out of \[n\] identical objects is 1.   (5) Total number of selections of zero or more objects from \[n\] identical objects is \[n+1\].   (6) The number of selections taking at least one out of \[{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+......+{{a}_{n}}\]+ k objects, where \[{{a}_{1}}\] are alike (of one kind), \[{{a}_{2}}\] are alike (of second kind) and so on......\[{{a}_{n}}\] are alike (of nth kind) and k are distinct   \[=[({{a}_{1}}+1)\,({{a}_{2}}+1)\,({{a}_{3}}+1).......({{a}_{n}}+1)]\,{{2}^{k}}-1\].

The number of combinations (selections or groups) that can be formed from \[n\] different objects taken \[r(0\le r\le n)\] at a time is \[^{n}{{C}_{r}}=\frac{n\,!}{r\,!(n-r)\,!}\]. Also \[^{n}{{C}_{r}}={{\,}^{n}}{{C}_{n-r}}\].   Let the total number of selections (or groups) \[=x\]. Each group contains \[r\] objects, which can be arranged in \[r\,\,!\] ways. Hence the number of arrangements of \[r\] objects \[=x\times \,(r!)\]. But the number of arrangements \[=\,{{\,}^{n}}{{P}_{r}}\].   \[\Rightarrow \,x(r!)={{\,}^{n}}{{P}_{r}}\Rightarrow x=\frac{^{n}{{P}_{r}}}{r\,!}\Rightarrow x=\frac{n!}{r\,!\,(n-r)\,!}{{=}^{n}}{{C}_{r}}\].