Important Results

1. Sum of digits in the unit place of all numbers formed by $a _{1}, a _{2} \ldots \ldots \ldots a _{n}$ taken all at a time is given by $(n-1) !\left(a _{1}+a _{2}+\ldots . . a _{n}\right)$ if repetition of digits is not allowe(d).

2. Sum of all the numbers which can be formed using the digits $\mathrm{a} _{1}, \mathrm{a} _{2} \ldots \ldots \ldots \mathrm{a} _{\mathrm{n}}$ (repetition not allowed)

$ \begin{aligned} & =\quad(\mathrm{n}-1) !\left(\mathrm{a} _{1}+\mathrm{a} _{2}+\ldots \ldots+\mathrm{a} _{\mathrm{n}}\right) \frac{\left(10^{\mathrm{n}}-1\right)}{9} \\ & =\quad(\mathrm{n}-1) !(\text { sum of digits) }(\underbrace{1 \ldots \ldots 1} _{\mathrm{n} \text { times }}) \end{aligned} $

Number of integral solutions of linear equations and unequations (Multinomial Theorem)

3. (i). Total number of non negative integral solutions of $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots \ldots \mathrm{x} _{\mathrm{r}}=\mathrm{n}$ is ${ }^{\mathrm{n}+\mathrm{r}-1} \mathrm{C} _{\mathrm{r}-1}$ Total number of positive integral solutions of $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots \ldots . \mathrm{x} _{\mathrm{r}}=\mathrm{n}$ is ${ }^{\mathrm{n}-1} \mathrm{C} _{\mathrm{r}-1}$

(ii). In order to solve inequations of the form $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots . .+\mathrm{x} _{\mathrm{r}} \leq \mathrm{n}$, we introduce artificial (dummy) variable $\mathrm{x} _{\mathrm{r}+1}$ such that $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots \ldots .+\mathrm{x} _{\mathrm{r}}+\mathrm{x} _{\mathrm{r}+1}=\mathrm{n}$ where $\mathrm{x} _{\mathrm{r}+1} \geq 0$.

Number of solutions of this equation are same as the number of solutions of inequation $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots . \mathrm{x} _{\mathrm{r}} \leq \mathrm{n}$.

(iii). Number of solutions of $\alpha+2 \beta+3 \gamma+\ldots . .+\mathrm{q} \theta=\mathrm{n}$ is

$\left\{\begin{array}{l}\text { coefficient of } x^{n} \text { in }(1-x)^{-1}\left(1-x^{2}\right)^{-1}\left(1-x^{3}\right)^{-1} \ldots \ldots .\left(1-x^{q}\right)^{-1} \text { if zero is included } \\ \text { coefficient of } x^{n} \text { in } x^{1+2+\ldots \ldots+q}(1-x)^{-1}\left(1-x^{2}\right)^{-1} \ldots \ldots .\left(1-x^{q}\right)^{-1} \text { if zero is not included }\end{array}\right.$

Number and sum of divisors

4. Let $\mathrm{N}=\mathrm{a}^{\mathrm{p}} \mathrm{b}^{\mathrm{q}} \mathrm{c}^{\mathrm{r}}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are primes & $\mathrm{p}, \mathrm{q}, \mathrm{r} \in \mathrm{Z}^{+}$.

(i). Number of divisors of $\mathrm{N}=(\mathrm{p}+1)(\mathrm{q}+1)(\mathrm{r}+1)$

Sum of divisors of $\mathrm{N}=\left(1+\mathrm{a}+\mathrm{a}^{2}+\ldots . \mathrm{a}^{\mathrm{p}}\right)\left(1+\mathrm{b}+\mathrm{b}^{2}+\ldots \mathrm{b}^{\mathrm{q}}\right)\left(1+\mathrm{c}+\mathrm{c}^{2}+\ldots . \mathrm{c}^{\mathrm{r}}\right)$

(ii). Number of ways in which $\mathrm{N}$ can be resolved as a product of two factors is

$ \left\{\begin{array}{l} \frac{1}{2}(a+1)(b+1)(c+1) \text { if } N \text { is not perfect square } \\ \left\{\frac{1}{2}(a+1)(b+1)(c+1)+1\right\} \text { if } N \text { is a perfect square } \end{array}\right\} $

5. Number of ways in which a composite number $\mathrm{N}$ can be resolved into two factors which are relatively prime (or co prime) to each other is $2^{\mathrm{n}-1}$ where $\mathrm{n}$ is the number of different prime factors is $\mathrm{N}$.

Divisibility

6. Condition for divisibility of a number

A number abcde will be divisible

1. by 4 if $2 \mathrm{~d}+\mathrm{e}$ is divisible by 4

2. by 8 if $4 \mathrm{c}+2 \mathrm{~d}+\mathrm{e}$ is divisible by 8

3. by 3 if $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}$ is divisible by 3

4. by 9 if $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}$ is divisible by 9

5. by 5 if $\mathrm{e}=0$ or 5

6. by 11 if $\underbrace{\mathrm{a}+\mathrm{c}+\mathrm{e}} _{\begin{array}{c}\text { Sum of digits } \\ \text { at odd places }\end{array}}-\underset{\begin{array}{c}\text { sur of digit } \\ \text { at even places }\end{array}}{\mathrm{b}+\mathrm{d}}$ in divisible by 11

7. by 6 if $\mathrm{e}=$ even and $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}$ is divisible by 3

8. by 18 if $\mathrm{e}=$ even and $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}$ is divisible by 9

Solved examples

1. Ten different letters are printed round a circle. The number of different ways in which we can select three letters so that no two of them are consecutive is

(a). 26

(b). 50

(c). 56

(d). 72

2. The numberof triangles whose vertices are the vertices of an octagon but none of whose sides happen to come from the sides of octagon is

3. The maximum number of points in which 4 circles and 4 straight lines intersect is

(a). 26

(b). 50

(c). 56

(d). 72

4. In a plane two families of lines are given by $y=x+r$ and $(y=-x+p)$ where $r \in\{0,1,2,3,4\}$ and $p \in\{0,1,2, \ldots \ldots, 9\}$. The number of squares of diagonals of length 3 units formed by these lines is

(a). 36

(b). 24

(c). 20

(d). none of these

5. The total number of words that can be made using letters of the word CALCULATE so that each word starts and ends with a consonant is

(a). $\frac{5.7 !}{2}$

(b). $\frac{3.7 !}{2}$

(c). $2.7 !$

(d). none of these

6. The number of 5 digit numbers of different digints in which middle digits is the largest is

(a). $ \sum _{\mathrm{n}=4}^{9} \mathrm{P} _{4}$

(b). $33(3 !)$

(c). $30(3 !)$

(d). none of these

Practice questions

1. Last digit of $(1 !+2 !+\ldots . .+2005 !)^{500}$ is

(a). 9

(b). 2

(c). 7

(d). 1

2. Number of integral solutions of $x+y+z=0$ with $x \geq-5, y \geq-5, z \geq-5$ is

(a). 134

(b). 136

(c). 138

(d). 140

3. Let $\mathrm{x} _{1}, \mathrm{x} _{2} \ldots, \mathrm{x} _{\mathrm{k}}$ be divisors of positive integer $\mathrm{n}$ (excluding 1 & $\mathrm{n}$ ). If $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots .+\mathrm{x} _{\mathrm{k}}=75$, then $\sum _{\mathrm{i}=1}^{\mathrm{k}} \frac{1}{\mathrm{x} _{\mathrm{i}}}$ is equal to

(a). $\frac{75}{\mathrm{n}^{2}}$

(b). $\frac{75}{n}$

(c). $\frac{75}{\mathrm{k}}$

(d). none of these

4. The total number of ways in which $\mathrm{n}^{2}$ number of identical balls can be put in $\mathrm{n}$ numbered boxes $(1,2,3, \ldots, n)$ such that $\mathrm{i}^{\text {th }}$ box contains at least $\mathrm{i}$ number of balls is

(a). ${ }^{n^{2}} C _{n-1}$

(b). ${ }^{n^{2}-1} C _{n-1}$

(c). $ \frac{n^{2}+n-2}{2} C _{n-1}$

(d). none of these

5. Total number of positive integral solutions of $15<\mathrm{x} _{1}+\mathrm{x} _{2}+\mathrm{x} _{3} \leq 20$ is

(a). 685

(b). 785

(c). 1125

(d). none of these

6. If $n$ is selected from the set $\{1,2,3 \ldots .10\}$ and the number $2^{n}+3^{n}+5^{n}$ is forme(d). Total number of ways of selecting $n$ so that the formed number is divisible by 4 is equal to

(a). 50

(b). 49

(c). 48

(d). none of these

7. $A$ is a set containing $n$ different elements. A subset $P$ of $A$ is chosen. The set $A$ is reconstructed by replacing the elements of $\mathrm{P}$. A subset $\mathrm{Q}$ of $\mathrm{A}$ is again chosen. The number of ways of choosing $\mathrm{P}$ and $\mathrm{Q}$ so that $\mathrm{P} \cap \mathrm{Q}$ contains exactly two elements is

(a). ${ }^{n} C _{3} \times 2^{n}$

(b). ${ }^{n} C _{2} \times 3^{n-2}$

(c). $3^{\mathrm{n}-2}$

(d). none of these

8. The number of three digit numbers of the form $x y z$ such that $x<y$ and $z \leq y$ is

(a). 276

(b). 285

(c). 240

(d). 244

9. Number of ordered triplets $(x, y, z)$ such that $x, y, z$ are primes and $x^{y}+1=z$ is

(a). 0

(b). 1

(c). 3

(d). none of these

10. Read the passage and answer the following questions:-

Suppose a lot of $n$ objects having $n _{1}$ objects one kind, $n _{2}$ objects of second kind, $n _{3}$ objects of third kind,…, $\mathrm{n} _{\mathrm{k}}$ objects of $\mathrm{k}^{\text {th }}$ kind satisfying the condition $\mathrm{n} _{1}+\mathrm{n} _{2}+\ldots+\mathrm{n} _{\mathrm{k}}=\mathrm{n}$, then the number of possible arrangements / permutations of $m$ objects out of this lot is the coefficient of $\mathrm{x}^{\mathrm{m}}$ in the expansion of $m ! \prod\left\{\sum _{\lambda=0}^{n i} \frac{x \lambda}{\lambda i}\right\}$

i. The number of permutations of the letters of the word AGAIN taken three at a time is

(a). 48

(b). 24

(c). 36

(d). 33

ii. The number of permutations of the letters of the word EXAMINATION taken 4 at a time is

(a). 136

(b). 2454

(c). 2266

(d). none of these

iii. The number of permutations of the letters of the word EXERCISES taken 5 at a time is

(a). 2250

(b). 30240

(c). 226960

(d). none of these

iv. The number of ways in which an arrangement of 4 letters of the word PROPORTION can be made is

(a). 700

(b). 750

(c). 758

(d). none of these

v. The number of permutations of the letters of the word SURITI taken 4 at a time is

(a). 360

(b). 240

(c). 216

(d). none of these