Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Number of Divisions (Applications)

Let N=p^{q}\cdot q ^{b}\cdot r^c \cdots , where P, q, r,\cdots are distinct primes & a,b,c,\cdots are natural numbers, then

  1. The total no. of divisors of N including 1\& \ N is (a+1)(b+1)(c+1)\cdots.
  2. The sum of divisors is (p^0+p^1+p^2+\cdots p^a)(ql^0+q^1+q^2+\cdots q^b)(r^0+r^1+r^2+\cdots r^l)\cdots
  3. Number of ways in which N can be resolved as a product of two factors is

 =\begin{cases} \dfrac{1}{2}(a+1)(b+1)(c+1)\cdots, & \mbox{ if } N \mbox{ is not a perfect square.}  \vspace*{0.18cm}\\  \dfrac{1}{2}[(a+1)(b+1)(c+1)\cdots +1], & \mbox{ if } N \mbox{ is a perfect square.}  \end{cases}

Distribution of ‘K’ balls into ‘n’ boxes : (K\neq n)

\rightarrow\ n^K : No restrictions.

\rightarrow\ ^{n}P_{K} : Each box can contain at most one ball (\leq 1).

\rightarrow\ S(K,n)\times n! : each ball can contain at least one ball (\geq 1).

\rightarrow\ \begin{array}{ll}  \begin{cases} ^{n}P_{n}, & \mbox{ if } K=n \vspace*{0.18cm}\\  0, & \mbox{ if } K\neq n\\  \end{cases}  &  \bigg( \mbox{Each box can contain exactly one ball (=1)} \bigg)  \end{array}

In the above 4 formulas n,\ K are distincts.

Distribution of ‘K’ identical balls into ‘n’ distinct boxes

\rightarrow\ ^{(K+n-1)}C_{(n-1)} : No restriction.

\rightarrow\ ^{n}C_{K} : Each box contain at most one ball (\leq 1).

\rightarrow\ ^{(K-1)}C_{(n-1)}: Each box contain at least one ball (\geq 1).

\begin{array}{ll}  \rightarrow\  \begin{cases} 1, & \mbox{ if } K=n \vspace*{0.18cm}\\  0, & \mbox{ if } K\neq n\\  \end{cases}  &  \bigg( \mbox{Each box can contain exactly one ball (=1)} \bigg)  \end{array}

Distribution of ‘K’ identical balls into ‘n’ identical boxes

\rightarrow \ \displaystyle \sum^{n}_{i=1}P(K,i) : No restriction.

\rightarrow\ \begin{array}{ll}  \begin{cases} 1, & \mbox{ if } K\leq n \vspace*{0.18cm}\\  0, & \mbox{ if } K> n\\  \end{cases}  &  \bigg( \mbox{Each box contain at most one ball }(\leq 1) \bigg)  \end{array}

\rightarrow\  P(K,n) : Each box contain at least one ball (\geq 1).

\rightarrow\ \begin{array}{ll}  \begin{cases} 1, & \mbox{ if } K= n \vspace*{0.18cm}\\  0, & \mbox{ if } K\neq n\\  \end{cases}  &  \bigg( \mbox{ Exactly one ball }(= 1) \bigg)  \end{array}

Distribution of ‘K’ distinct balls into ‘n’ identical boxes

\rightarrow \ \displaystyle \sum^{n}_{i=1}S(K,i) : No restriction.

\rightarrow\ \begin{array}{ll}  \begin{cases} 1, & \mbox{ if } K\leq n \vspace*{0.18cm}\\  0, & \mbox{ if } K> n\\  \end{cases}  &  \bigg( \mbox{  at most one ball }(\leq 1) \bigg)  \end{array}

\rightarrow\  S(K,n) : Each box contain at least one ball (\geq 1).

\rightarrow\ \begin{array}{ll}  \begin{cases} 1, & \mbox{ if } K= n \vspace*{0.18cm}\\  0, & \mbox{ if } K\neq n\\  \end{cases}  &  \bigg( \mbox{ Each box contain exactly one ball }(= 1) \bigg)  \end{array}

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