PERMUTATIONS
FACTORIALS
The continued product of Iⁿ n natural number is called the "n!"
n factorial
ie; n!=1*2*3*.......
0!=1
n!=1*2*3........(n-2)(n-1)n
n!=(n-1)!n
factorial is defined only for whole numbers
PERMUTATIONS
A permutation of a set of elements is an arrangement of (all or part) elements in same order
Eg: n=3
{a, b, c}
3P(3) abc
acb
bac
bca
cab
cba
The number of permutationn of n things taken r at a time is the no. of ways of arranging n things taken r at atime and denoted by P(nr)=ⁿPᵣ=nᵖr
This is equivalent to the number of ways of filling up r vacent spaces in a line with n different objects
Theorem
If 0≤r≤n
then ⁿPᵣ = (n)(n-1)(n-2)............(n-(r-1))
=(n)(n-1)(n-2)..............(n-r+1)
Note o≤r≤n
npᵣ=n!/(n-r)!
RHS= n!/(n-r)=(1*2*3.......(n-r-1)(n-r)(n-r+1).............(n-2)(n-1)n)/(1*2*3.......(n-r-1)(n-r))
=n(n-1)(n-2)........(n-r+1)
RHS
Note
(i) npₙ=n!/(n-n)!
=n!/0!
=n!
(ii) npₙ=n!/(n-0)!
= n!/n!
=1
(ii) np₁=n!/(n-1)!=n
1) Find 10p₂
2) Find r if 10pᵣ=2*9pᵣ
3)Solve for n if np₂=30
4)Find n if (n-1)p₃: np₄ = 1:9
5) Prove that
npᵣ=ⁿ⁻¹pᵣ+r ⁽ⁿ⁻¹⁾p₍ᵣ₋₁₎
6)In how many ways can 4 boys and 3 girls be seated in a row containing 7 seats seats, if the girls and boys must alternate?
7) Find the no of different ways in which 6 boys and 4 girls may be arranged in a row such that no two girl sit together?
8)Find the number of 8 letter words formed from the letters of word " EQUATION "If each word is to start with wovel?
9)How many 4 letter word with or without meaning, can be formed out of letters of the word
"LOGARITHMS"
If repitation of letters not allowed
10)How many even numbers of four digit can be formed with figures 2,4,5,7,8 no digit being used more than once
11) If the letters of word "RAJESH" be permuted and arranged as in a dictionary.
Find the Rank of RAJESH
......................................
The number of permutations of n different objects taken r at a time when repetitions are allowed is nⁿ
.....................................
12)Find the number of ways of answering five objective type question, each question having 4 chances
13) There are 6 multiple choice questions.
How many sequence of answers are possible if the first three questions have 4 choices each and the next three has 5 choices
14)Find the number of 3 letter word formed by the letters of the word
"NUMBER"
(i)If repetitions are not allowed
(ii) If repetitions are not allowed
15) In how many ways 6 letters can be posted in 3 letter boxes
16)In how many ways 5 rings can be worn in 4 fingers
ans)4⁵
PERMUTATIONS WHEN ALL THE OBJECTS ARE NOT DISTINCT
THEORM
The number of permutations of n objects where P objects are of the same type and rest are different
ie: n!/p!
theorm
The number of permutations of n objects where P₁, objects are of one kind
The number of permutations P₂ are of second kind etc, Pₐ are of k times and the the rest.
If any are at different kind is
n!/P₁!P₂!P₃!..... Pₐ!
