Permutation and combination important formulas and facts
Factorial:Let
n be a positive integer.Then factorial of n denoted by n! is defined as
n!=1*2*3*……………..(n-2)*(n-1)*n
Note:0!=1
Permutation:
Different arrangement of a given number of things by taking some or all at a
time.
Example:All
arrangements made with letters a,b,c by taking two at a time are
ab,ba,bc,cb,ca,ac.
All
arrangements made with letters a,b,c by taking all at a time are
abc,bca,cab,cba,acb,bac.
Number of permutation:Total
number of possible arrangements(permutation) of n things, taken r at a time, is
given by:
Example 1.Arrangement
of 3 items taken 2 at a time
Example2. Arrangement
of 4 items taken all at a time
Permutation
:Important point to note
-Number of
all permutation of n things all at a time is n!
-If there
are n objects ,m numbers are alike
Then number
of permutations of these objects is :
-If there
are n objects ,p1 numbers are alike of one kind,p2 objects are alike of another
kind,p3 are alike of third kind and so on and pr are alike of rth kind,such
that p1+p2+p3+…………pr=n
Then number
of permutations of these objects is :
Combination: Each of
different groups or selection which can be formed by taking some or all of a
number of object, is called a combination.
Suppose we want to
select two students from a group of three students namely A,B and C.Then,
possible selections are AB,BC and CA.
Note AB and BA
represents same selection. But in permutation/arrangement AB and BA represents
two different arrangements.
If we want to select
‘all at a time ‘, then there is only one possibility ABC.
Number of
combinations:The number of all combination of n things, taken r at a time is:
Example:
Practice
questions
1.How many
arrangements are possible using all the letters of the word BIHAR?
Solution:The
word BIHAR contains 5 different letters.
Required
number of arrangements = 5P5=5!=120
2.How many
word can be formed using all the letters of the word DAUGHTER so that vowels
always come together?
There are 8
different letters in the given word,but it is given that vowels should come
together.Treat vowels as a single entity ‘AUE’
So total
number of letters is 5 different letters+ vowel entity=6
Total
arrangement possible using 6 letters=6P6=6!=720
Vowels can
be arranged in 3P3 ways. 3P3=6
Total number
of arrangement=720*6=4320