Permutation-and-combination Formula Sheet

Permutation refers to the arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is denoted by $n_{P_r}$ and calculated as $\frac{n!}{(n-r)!}.$
Combination refers to the selection of objects without considering the order. The number of combinations of n objects taken r at a time is denoted by $n_{C_r}$ and calculated as $\frac{n!}{r!(n-r)!}.$

Neetesh Kumar | June 04, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Fundamental Principle of Counting
• 2. Factorial
• 3. Permutation
• 4. Combination
• 5. Divisors
• 6. Division into Groups and Distribution
• 7. Derangement
• 8. Important Result

1. Fundamental Principle of Counting:

If an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways, then the total number of different ways of:
(a) Simultaneous occurrence of both events in a definite order is $m \times n$. This can be extended to several events (known as the multiplication principle).
(b) The happening of exactly one of the events is $m + n$ (known as the addition principle).

2. Factorial:

It is a Useful Notation:
$n! = n (n – 1) (n – 2) ...... 3. 2. 1;$

$n! = n. (n – 1)!$ where $n \in W$

$0! = 1! = 1$

$(2n)! = 2n . n! [1. 3. 5. 7........(2n – 1)]$

Note:

(i) Factorial of negative integers is not defined.

(ii) Let $p$ be a prime number and $n$ be a positive integer,
then exponent of $p$ in $n!$ is denoted by $E_p (n!)$ and is given by

$E_p (n!) = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + ....$

3. Permutation:

(a) $n_{P_r}$ denotes the number of permutations (arrangements) of $n$ different things, taken $r$ at a time ($n \in N, r \in W, n \geq r$)

$n_{P_r} = n (n – 1) (n – 2) ............. (n – r + 1) = \frac{n!}{(n - r)!}$

(b) The number of permutations of $n$ things taken all at a time when $p$ of them are similar of one type, $q$ of them are similar of a second type, $r$ of them are similar of a third type and the remaining $n – (p + q+ r)$ are all different is: $\frac{n!}{p! q! r!}$

(c) The number of permutations of $n$ different objects taken $r$ at a time, when a particular object is always to be included is $r! ({n–1}_{C_{r–1}})$

(d) The number of permutations of $n$ different objects taken $r$ at a time, when repetition be allowed any number of times is $n \times n \times n \times .............. \text{r times} = n^r$

(e)

• The number of circular permutations of $n$ different things taken all at a time is; $(n – 1)! = \frac{n!}{n}$.

• If clockwise & anti-clockwise circular permutations are considered to be the same, then it is $\frac{(n - 1)!}{2}$.

• The number of circular permutations of $n$ different things taking $r$ at a time distinguishing clockwise & anticlockwise arrangement is $\frac{n_{P_r}}{r}$.

4. Combination:

(a) $n_{C_r}$ denotes the number of combinations (selections) of $n$ different things taken $r$ at a time, and $n_{C_r} = \frac{n!}{r!(n - r)!}$ where $r \leq n$, $n \in N$ and $r \in W$. $n_{C_r}$ is also denoted by $\binom{n}{r}$ or $C (n, r)$.

(b) The number of combinations of $n$ different things taken $r$ at a time:

• when $p$ particular things are always to be included = ${(n-p)}_{C_{(r-p)}}$

• when $p$ particular things are always to be excluded = ${(n-p)}_{C_{r}}$

• when $p$ particular things are always to be included and $q$ particular things are to be excluded = ${(n-p-q)}_{C_{(r-p)}}$

(c) Given $n$ different objects, the number of ways of selecting at least one of them is $n_{C_1} + n_{C_2} + n_{C_3} +........+ n_{C_n} = 2^n – 1$.

This can also be stated as the number of non-empty combinations of $n$ distinct things.

(d)

• The total number of ways in which it is possible to make a selection by taking some or all out of $p + q + r +......$ things, where $p$ are alike of one kind, $q$ alike of a second kind, $r$ alike of a third kind & so on is given by: $(p + 1) (q + 1) (r + 1).........–1$.

• The total number of ways of selecting one or more things from $p$ identical things of one kind, $q$ identical things of a second kind, $r$ identical things of a third kind and $n$ different things is $(p + 1) (q + 1) (r + 1) 2^n – 1$.

5. Divisors:

Let $N = p^a q^b r^c .......$ where $p, q, r........$ are distinct primes & $a, b, c.......$ are natural numbers then:

(a) The total number of divisors of $N$ including 1 & $N$ is $= (a + 1) (b + 1) (c + 1).......$

(b) The sum of these divisors is $(p^0 + p^1 + p^2 + ....+ p^a) (q^0 + q^1 + q^2 + ....+ q^b) (r^0 + r^1 + r^2 + ....+ r^c)...$

(c) The number of ways in which $N$ can be resolved as a product of two factors is
= $\left\lfloor \frac{(a + 1) (b + 1) (c + 1).......}{2} \right\rfloor$ if $N$ is not a perfect square.

$= \left\lfloor \frac{(a + 1) (b + 1) (c + 1)....... + 1}{2} \right\rfloor$ if $N$ is a perfect square.

(d) The number of ways in which a composite number $N$ can be resolved into two factors that are relatively prime (or coprime) to each other is equal to $2^{n-1}$ where $n$ is the number of different prime factors in $N$.

6. The division into Groups and Distribution:

(a)

• The number of ways in which $(m + n)$ different things can be divided into two groups containing $m$ & $n$ things respectively is: $\frac{(m + n)!}{m! n!}$ $(m \neq n)$.

• If $m = n$, then number of ways in which $2n$ distinct objects can be divided into two equal groups is $\frac{(2n)!}{n! n! 2!}$; as in any one way, it is possible to interchange the two groups without obtaining a new distribution.

• If $2n$ things are to be divided equally between two persons, then the number of ways = $\frac{(2n)!}{2! n! n! (2!)}$.

(b)

• Number of ways in which $(m + n + p)$ different things can be divided into three groups containing m, n & p things respectively is $\frac{(m + n + p)!}{m! n! p!}$, $m \neq n \neq p$.

• If $m = n = p$ then the number of such grouping is $\frac{(3n)!}{n! n! n! 3!}$.

• If $3n$ things are to be divided equally among three people, then the number of ways in which it can be done is $\frac{(3n)!}{(n!)^3}$.

(c) In general, the number of ways of dividing $n$ distinct objects into $l$ groups containing $p$ objects each, $m$ groups containing $q$ objects each is equal to $\frac{n!}{(p!)^l (q!)^m (l! m!)}$ Here $lp + mq = n$

(d) Number of ways in which $n$ distinct things can be distributed to $p$ persons if there is no restriction to the number of things received by them is $p^n$

(e) Number of ways $n$ identical things may be distributed among $p$ persons if each person receives none, one or more things is ${(n+p–1)}_{C_n}$.

7. Derangement:

The number of ways in which $n$ letters can be placed in $n$ directed envelopes so that no letter goes into its own envelope is

$= n! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - .... + (-1)^n \frac{1}{n!} \right]$

8. Important Result:

(a) The number of rectangles of any size in a square of size $n \times n$ is $\sum_{r=1}^{n} r^3$ & number of squares of any size is $\sum_{r=1}^{n} r^2$.

(b) The number of rectangles of any size in a rectangle of size $n \times p$ ($n < p$) is $\frac{np (n + 1) (p + 1)}{4}$ & number of squares of any size is $\sum_{r=1}^{n} (n + 1 - r) (p + 1 - r)$.

(c) If there are $n$ points in a plane of which $m (<n)$ are collinear:

(i) Total number of lines obtained by joining these points is $n_{C_2} – m_{C_2} + 1$

(ii) Total number of different triangles is $n_{P_3} – m_{P_3}$

(d) Maximum number of points of intersection of $n$ circles is $n_{P_2}$ and $n$ lines is $n_{C_2}$.

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