PERMUTATIONS & COMBINATIONS
Factorial: n! = n (n – 1) (n – 2) …….3.2.1. (0! = 1)
Permutation: different arrangements of a given number of things by taking
some or all at a time
E.g.: all permutations made with the letters a, b, c by taking two at a
time are ab, bc, ac, ca, bc, cb
Number of permutations of n things, taken r at a time:
nP = n (n – 1) (n – 2) …….(n – r + 1) = n! / (n – r) !
r
Number of all permutations of n things, taken all at a time = n !
PERMUTATIONS & COMBINATIONS
If out of n things, p are exactly alike of one kind, q exactly alike of second kind
and r exactly alike of third kind and the rest are different, then the number of
permutations of n things taken all at a time = n! / p! q! r!
Combinations: each of the different groups/selections which can be formed by
taking some or all of a number of objects, is called a combination.
E.g.: all combinations made with the letters a, b, c by taking two at a
time are ab, bc and ca
Number of all combinations of n things, taken r at a time is:
nC = n (n – 1) (n – 2) …… to r factors / r! = n! / (r !) (n – r)!
r
PERMUTATIONS & COMBINATIONS
nC =1 and nC =1
n 0
nC = nC n - r
r
PERMUTATIONS & COMBINATIONS
Q.1) How many words can be formed by using all letters of the
word ‘SURAT’?
a) 60
b) 120
c) 30
d) 90
PERMUTATIONS & COMBINATIONS
Answer - B
PERMUTATIONS & COMBINATIONS
Q.2) How many words can be formed by using all letters of the
word ‘DAUGHTER’ so that the vowels always come together?
a) 6720
b) 720
c) 5040
d) 4320
PERMUTATIONS & COMBINATIONS
Answer - D
PERMUTATIONS & COMBINATIONS
Q.3) How many words can be formed by using all letters of the
word ‘DRONE’ so that the vowels are never together?
a) 120
b) 48
c) 72
d) 60
PERMUTATIONS & COMBINATIONS
Answer - C
PERMUTATIONS & COMBINATIONS
Q.4) In how many ways a group of 5 students can be selected from 6
boys and 5 girls, consisting of 3 boys and 2 girls?
a) 200
b) 30
c) 100
d) 120
PERMUTATIONS & COMBINATIONS
Answer - A
PERMUTATIONS & COMBINATIONS
Q.5) There are 4 flags of different colours. How many different signals
can be given by taking any number of flags at a time?
a) 48
b) 64
c) 24
d) 256
PERMUTATIONS & COMBINATIONS
Answer - B
PERMUTATIONS & COMBINATIONS
Q.6) How many arrangements can be made out of the letters of the
word ‘ENGINEERING’?
a) 23100
b) 69300
c) 277200
d) 92400
PERMUTATIONS & COMBINATIONS
Answer - C
PERMUTATIONS & COMBINATIONS
Q.7) From a group of 7 men and 6 women, five persons are to be selected to
form a committee so that at least 3 men are there on the committee. In how
many ways can it be done?
a) 756
b) 645
c) 564
d) 735
PERMUTATIONS & COMBINATIONS
Answer - A
PERMUTATIONS & COMBINATIONS
Q.8) A box contains 2 white balls, 3 black balls and 4 red balls. In how many
ways can 3 balls be drawn from the box, if at least one black ball is to be
included in the draw?
a) 96
b) 64
c) 48
d) 32
PERMUTATIONS & COMBINATIONS
Answer - B
PERMUTATIONS & COMBINATIONS
Q.9) How many 3-didgit numbers can be formed from the digits 2, 3, 5, 6, 7 and
9, which are divisible by 5 and none of the digits is repeated?
a) 5
b) 10
c) 15
d) 20
PERMUTATIONS & COMBINATIONS
Answer - D
PERMUTATIONS & COMBINATIONS
Q.10) If there are 10 persons in a party, and each person shake hands with all
the persons in the party, then how many hand shakes took place in the party?
a) 75
b) 60
c) 45
d) 90
PERMUTATIONS & COMBINATIONS
Answer - C
