Permutations and Combinations

Why?

Why do one need permutations and combinations in one’s real life?

Ø To get the number of possibilities to arrange or select a few things from a given set of things, using these rather than a mere manual tabulation.

Permutation:

The permutation is defined as each of several possible ways in which a set or number of things can be ordered or arranged.

Ø Mathematically given by nPr =

Combination:

The combination is defined as a joining or merging of different parts or qualities in which the component elements are individually distinct.

Ø Mathematically given by nCr =

Now let us see the physical interpretation of each:

1. The permutation can be interpreted as the number of arrangements or sequences you can make with a given set of characters.

Eg: Suppose there are few buckets Red, Blue, Green, Yellow and A, B, C, D. Now, how many sequences can be done such that each letter sits in each bucket.

Answer is

				sl no
A	B	C	D	1
A	B	D	C	2
A	C	B	D	3
A	C	D	B	4
A	D	B	C	5
A	D	C	B	6
B	A	C	D	7
B	A	D	C	8
B	C	A	D	9
B	C	D	A	10
B	D	A	C	11
B	D	C	A	12
C	A	B	D	13
C	A	D	B	14
C	B	A	D	15
C	B	D	A	16
C	D	B	A	17
C	D	A	B	18
D	A	B	C	19
D	A	C	B	20
D	B	A	C	21
D	B	C	A	22
D	C	B	A	23
D	C	A	B	24

i.e. total of 24 different possible ways, but isn’t it too tedious?
Now, number of buckets = 4 = n

Number of buckets = 4 = r

4p4 =

= 4!/1

= 4 × 3 × 2 × 1

= 24

which is as same as the above answer.

2. The combination can be interpreted as the number of selections can be done from the total number of characters given.

Eg: Say A, B, C, D are the given characters and said to select 2 characters on random. What are the different possibilities in this case?

The answer is either we can take

(A&B) or (A&C) or (A&D) or (B&C) or (B&D) or (C&D)

Now, number of given char = 4 = n

number of selections from given = 2 = r

4p4 =

=

= × 2 × 1

= 6

which is as same as the above answer.