DERANGEMENTS
Derangement means destroy the arrangement i.e.,
rearranging the objects in such a way that no object
remains at its original place.
z If n distinct things are arranged in a row, then
number of ways in which they can be deranged
such that none of them occupies its original place
= n! – nC 1 (n – 1)! + nC 2 (n – 2)! – nC 3 (n – 3)!
- ... + (–1)n nCn⋅ 0!
=n⎝⎛⎜ − + − ++()− ⎞⎠⎟
n
! n
!!!!1 1
11
21
3" 1 1z If r(0 ≤ r ≤ n) objects occupy the places assigned
to them i.e., their original places and none of the
remaining (n – r) objects occupies its original
place, then the number of such ways
= ⋅− −+ − ++−
−
⎧
⎨
⎩⎫
⎬
⎭nCnrr nr−
nr()!
!!!...... ( )
()!1 1
11
21
31 1PROBLEMS- n 1 and n 2 are four digit numbers. Total number of
ways of forming n 1 and n 2 so that n 2 can be subtracted
from n 1 without borrowing at any stage, is equal to
(a) (36) (55)^3 (b) (45) (55)^3
(c) (55)^4 (d) None of these - Total number of positive integral solutions of
15 < x 1 + x 2 + x 3 ≤ 20 is equal to
(a) 1125 (b) 1150
(c) 1245 (d) 685 - ‘n’ is selected from the set {1, 2, 3, ....., 100} and the
number 2n + 3n + 5n is formed. Total number of ways of
selecting ‘n’ so that the formed number is divisible by 4,
is equal to
(a) 50 (b) 49
(c) 48 (d) none of these - Total number of times, the digit ‘3’ will be written,
when the integers having less than 4 digits are listed, is
equal to.
(a) 300 (b) 271
(c) 298 (d) none of these - A variable name in certain computer language
must be either a alphabet or alphabet followed by a
digit. Total number of different variable names that can
exist in that language is equal to
(a) 280 (b) 240
(c) 286 (d) 80
6. Total number of ways of selecting two numbers
from the set {1, 2, 3, 4, ......., 3n} so that their sum is
divisible by 3, is equal to
(a)^2
2nn^2 −
(b)^3
2nn^2 −(c) 2n^2 – n (d) 3n^2 – n- The total number of ways of selecting 10 balls out
of an unlimited number of identical white, red and blue
balls, is equal to
(a)^12 C 2 (b)^12 C 3
(c)^10 C 2 (d)^10 C 3 - There are 10 persons among whom two are brothers.
The total number of ways in which these persons can be
seated around a round table so that exactly one person
sits between the brothers, is equal to
(a) (2!) (7!) (b) (2!) (8!)
(c) (3!) (7!) (d) (3!) (8!) - A library has ‘a’ copies of one book, ‘b’ copies each
of two books, ‘c’ copies each of three books and single
copy of ‘d’ books. The total number of ways in which
these books can be arranged in a shelf, is equal to
(a) ()!
!( !) ( !)
abcd
ab c+++ 23
23 (b)()!
!( !)( !)abcd
ab c+++ 23
233(c) ()!
(!)abcd
c+++ 23
3 (d)()!
!( !)( !)abcd
abc+++ 23
2- The number of numbers that are less than 1000
that can be formed using the digits 0, 1, 2, 3, 4, 5 such
that no digit is being repeated in the formed number, is
equal to
(a) 130 (b) 131 (c) 156 (d) 155 - Total number of six digit numbers that can be
formed, having the property that every succeeding digit
is greater than the preceding digit, is equal to
(a)^9 C 3 (b)^10 C 3 (c)^9 P 3 (d)^10 P 3 - If letters of the word ‘KUBER’ are written in all
possible orders and arranged as in a dictionary, then
rank of the word ‘KUBER’ will be
(a) 67 (b) 68 (c) 65 (d) 69 - In a chess tournament, all participants were to play
one game with the other. Two players fell ill after having
played 3 games each. If total number of games played
in the tournament is equal to 84, then total number of
participants in the beginning was equal to
(a) 10 (b) 15 (c) 12 (d) 14