Permutation and Combination

Permutation: The different arrangements of a given number or things by taking some or all at a time , are called Permutations.
 All permutations (or arrangements) made with the letters of a, b, c by taking two at a time are ( ab, ba, ac, ca, bc, cb )
All permutations made with the letters a, b, c, taking all at a time are: ( abc, acb, bac, bca, cab, cba ).

Number of Permutations: Number of all permutations of n things, taken r at a time, is given by:
n p r = n ( n – 1 ) ( n – 2 )……(n – r + 1 ) = n! / ( n – r ) !
Ex.6 p 2
= ( 6 x 5 ) = 30 .
Ex.7 p 3
= ( 7 x 6 x 5 ) = 210 ;
Note: Number of all permutation that is n things, we can take all at a time = n!


 In how many different ways the words ‘HOUSE’ can be arranged?
 The word HOUSE contains 5 letters.
Required number of word using formula 5p5 = 5! = (1 x 2 x 3 x 4 x 5) = 120.

In how many several way the word KOLKATA can be prepared so that as vowel always come together?
First of all we need to know how many vowel in this given word, here is three vowel that is O and A, A.
then we count the consonant that is four. Now we count the number of vowel as a single unit that is vowel O A, A count as single unit and add it with consonant so we have a four unit.
(4 consonant +three vowel as single unit) = 5 x 3(three vowel)
5! x 3! = 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 = 720 ways we prepared the word.

In how many several way the word HOUSE can be prepared so that as vowel always come together?
First of all we need to know how many vowel in this given word, here is three vowel that is E ,O and U.
then we count the consonant that is Two. Now we count the number of vowel as a single unit that is vowel E,O and U count as single unit and add it with consonant so we have a  three unit.
(2 consonant + three vowel as single unit ) = 3 x 3( three vowel)
3! x 3! = 1 x 2 x 3 x 1 x 2 x 3 = 36 ways we prepared the word.

 In how many several ways the word CAME can be arranged, that the vowels not come together?
First of all we need to know how many vowel in this given word, here is two vowel that is A and E.
then we count the consonant that is Two.Now we count the number of vowel as a single unit that is vowel A and E count as single unit and add it with consonant so we have a three unit.
(2 consonant + two vowel as single unit ) = 3 x 2( two vowel)
Note: If we Subtract total number ways from within vowel ways than we get the not come to-gather ways,
So, ( 4! – 3! x 2! ) = 12.

What is Combination?
Each of the different groups or selections which can be formed by taking some or all of a number of objects, is called a combination. In combination we can select only one item at time it based on selection on choice.
Suppose Ball , Marbel, etc.

Type 1: Suppose we want to select two out of three girls A, B, C. Then possible selection are AB, BC, and CA.
Note: that AB and BA represent the same selection.

Type 2: Suppose We select all three girls Like A, B, C. Then the selection is ABC.
Note: BAC, ABC, CAB are the same choice of selection.

Type 3: All the combinations formed by a, b, c, taking two at a time are ab, bc, ca.

Type4: The only combinations that can be formed  of three letters a, b, c taken all at a time is abc.

Type5: Various person of 2 out of four person A, B, C, D are : AB,  AC, AD, BC, BD, CD.

Type6: Note that ab and ba are two different permutations but they represent the same combination.
Number of Combinations: The number of all combinations of n things, taken r at a time is :
n C r = n! / r! ( n – r ) ! = n ( n – 1 ) ( n – 2 ) …….to r factors / r!.
Note: if n = r , nCr = 1 and n C 0 = 1.


Important Rule:
nCr = n C (n – r ). = 1.
 find the value of 16 C 13
 16 C ( 16 – 13 ) = 16 C 3 = (16 x 15 x 14 x 13!) / (3!)
                     = (16 x 15 x 14) / (3 x 2 x 1) = 560

Find out value of 10 P 3
  10 C 3= 10 x 9 x 8 / 3! = 10 x 9 x 8 / 3 x 2 x 1 = 120

Find the value of 50 C 50
= 50 C 50 = 1. [ n C n = 1]