Permutation
: Permutation means *arrangement* of
things. The word *arrangement *is used, if the
order of things *is considered*.

Combination: Combination
means *selection* of things. The word *selection*
is used, when the order of things has *no importance*.

**Example:**
Suppose we have to form a number of consisting of three
digits using the digits **1,2,3,4**, To form
this number the digits have to be *arranged*.
Different numbers will get formed depending upon the
order in which we arrange the digits. This is an example
of *Permutation*.

Now
suppose that we have to make a team of 11 players out of
20 players, This is an example of *combination*,
because the order of players in the team will not result
in a change in the team. No matter in which order we list
out the players the team will remain the same! For a
different team to be formed at least one player will have
to be changed.

Now let us look at two fundamental principles of counting:

Addition rule : If an experiment can be performed in ‘n’ ways, & another experiment can be performed in ‘m’ ways then either of the two experiments can be performed in (m+n) ways. This rule can be extended to any finite number of experiments.

Example:
Suppose there are 3 doors in a room, 2 on one side and 1
on other side. A man want to go out from the room.
Obviously he has ‘3’ options for it. He can
come out by door ‘A’ or door ‘B’ or
door ’C’.

Multiplication Rule : If a work can be done in m ways, another work can be done in ‘n’ ways, then both of the operations can be performed in m x n ways. It can be extended to any finite number of operations.

Example.: Suppose a man wants to cross-out a room, which has 2 doors on one side and 1 door on other site. He has 2 x 1 = 2 ways for it.

Factorial n : The product of first ‘n’ natural numbers is denoted by n!.

n! = n(n-1) (n-2) ………………..3.2.1.

Ex. 5! = 5 x 4 x 3 x 2 x 1 =120

** Note **
0! = 1

Proof n! =n, (n-1)!

Or (n-1)! = [n x (n-1)!]/n = n! /n

Putting n = 1, we have

O! = 1!/1

or 0 = 1

Permutation

Number of permutations of ‘n’ different things taken ‘r’ at a time is given by:-

^{n}P_{r}
=
n!/(n-r)!

**Proof**:
Say we have ‘n’ different things a_{1},
a_{2}……, a_{n}.

Clearly the first place can be filled up in ‘n’ ways. Number of things left after filling-up the first place = n-1

So the second-place can be filled-up in (n-1) ways. Now number of things left after filling-up the first and second places = n – 2

Now the third place can be filled-up in (n-2) ways.

Thus number of ways of filling-up first-place = n

Number of ways of filling-up second-place = n-1

Number of ways of filling-up third-place = n-2

Number of ways of filling-up r-th place = n – (r-1) = n-r+1

By multiplication – rule of counting, total no. of ways of filling up, first, second — rth-place together :-

n (n-1) (n-2) ———— (n-r+1)

Hence:

^{n}P_{r}
= n (n-1)(n-2) ————–(n-r+1)

= [n(n-1)(n-2)———-(n-r+1)] [(n-r)(n-r-1)—–3.2.1.] / [(n-r)(n-r-1)] —-3.2.1

^{n}P_{r} = n!/(n-r)!

Number of permutations of ‘n’ different things taken all at a time is given by:-

^{n}P_{n}
=
n!

Proof
:

Now we have ‘n’ objects, and n-places.

Number of ways of filling-up first-place = n

Number of ways of filling-up second-place = n-1

Number of ways of filling-up third-place = n-2

Number of ways of filling-up r-th place, i.e. last place =1

Number
of ways of filling-up first, second, — n th place

= n (n-1) (n-2) —— 2.1.

^{n}P_{n} = n!

Concept.

We have ^{n}P_{r}
= n!/n-r

Putting r = n, we have :-

^{n}P_{r} = n!
/ (n-r)

But ^{n}P_{n
}= n!

Clearly it is possible, only when n! = 1

Hence it is proof that 0! = 1

**Note**
: Factorial of negative-number is not defined. The
expression –3! has no meaning.

__Examples__

Q. How many different signals can be made by 5 flags from 8-flags of different colours?

Ans.
Number of ways taking 5 flags out of 8-flage = ^{8}P_{5}

= 8!/(8-5)!

= 8 x 7 x 6 x 5 x 4 = 6720

Q. How many words can be made by using the letters of the word “SIMPLETON” taken all at a time?

Ans. There are ‘9’ different letters of the word “SIMPLETON”

Number
of Permutations taking all the letters at a time = ^{9}P_{9}

= 9! = 362880.

Number of permutations of n-thing, taken all at a time, in which ‘P’ are of one type, ‘g’ of them are of second-type, ‘r’ of them are of third-type, and rest are all different is given by :-

n!/p! x q! x r!

Example: In how many ways can the letters of the word “Pre-University” be arranged?

13!/2! X 2! X 2!

Number of permutations of
n-things, taken ‘r’ at a time when each thing
can be repeated r-times is given by = n^{r}.

Proof.

Number of ways of filling-up first –place = n

Since repetition is allowed, so

Number of ways of filling-up second-place = n

Number of ways of filling-up third-place

Number of ways of filling-up r-th place = n

Hence total number of ways in which first, second —-r th, places can be filled-up

= n x n x n ————- r factors.

= n^{r}

Example: A child has 3 pocket and 4 coins. In how many ways can he put the coins in his pocket.

Ans. First coin can be put in 3 ways, similarly second, third and forth coins also can be put in 3 ways.

So total number of ways
= 3 x 3 x 3 x 3 = 3^{4}
= 81