Permutations and combinations

Fundamental principle of counting

Mohan has 3 pants and 2 shirts. How many different pairs of a pant and a shirt, can he dress up with?

There are 3 ways in which a pant can be chosen, because there are 3 pants available. Similarly, a shirt can be chosen in 2 ways. For every choice of a pant, there are 2 choices of a shirt. Therefore, there are 3 x 2=6 pairs of a pant and a shirt.

A boy wants to buy 3 different books from a book shop. In that shop there are 5 different books. He selects 3 books out of 5. Selection of 3 books out of 5 is a combination. He arranges these books in his shelf in different ways every day. Each way of the arrangement of books is a permutation.

• Permutation is an orderly arrangement of objects.
• Combination is mere selection of objects.

Permutation:

Definition: A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.

Examples:

1.  There are three boys: ASHRITH (A), BHARATH (B), and CHETAN(C). Arrange them in a row, two at a time. The arrangements are

There are 6 different ways of arrangements. These arrangements are called permutations.

Example 2: there are four different digits 2, 3, 4, 5. How many numbers can be formed using two digits at a time?

 From four different digits, twelve 2-digit numbers are formed.

From the above examples it is clear that for 1

• the number of all permutations of n distinct things taken r at a time is given
  

   by n(n-1)(n-2)…(n-)

• nPr=

• the number of all permutations of n distinct things, taken all at a time is nPn =n !.
• 0!=1
• nPn=nPn-1
• 1!=1
• The number of circular permutation of n things in which c lockwise and anticlockwise arrangements give rise to different permutation is (n-1)!.
• If there are n things and if the direction is not taken into considerat ion , the number of circular permutations is ½ (n-1)!

Combinations:

Example:

There are three boys: ASHRITH (A), BHARATH (B), and CHETAN(C). From these boys how many groups of two boys can be selected?

The selections are AB, AC and BC. So, 3 groups can be selected.

Note: Here AB and BA; AC and CA; BC and CB are not different selections

In general the number of combinations of n things taken r at a time is denoted by ⁿC_r.

Observe the following:

For 0 ≤ r ≤ n

• nCr=
• nCr=

• nCn=1
• nC0=1
• nCr=nCr-1
• ncr + nC(r-1)=(n+1)Cr