CST

Permutation & Combination

Session 2
Quantitative Aptitude
01

How many diagonals does a decagon have?

A 10C B 10C - 10
2 2

C 10P
2 D 10P
2 - 10

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01

How many diagonals does a decagon have?

A 10C B 10C - 10
2 2

C 10P
2 D 10P
2 - 10

Answer
01

• A decagon has 10 vertices

• The number of ways to connect these 10 vertices = 10C2 = 45
• To get the number of diagonals, subtract the number of sides from 45.
• Thus, number of diagonals = 45 – 10 = 35

Hints
02

In how many ways can you arrange the letters in the word DRAVID such
that the letter R comes before the Ds?

A 720 B 360

C 180 D 120

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02

In how many ways can you arrange the letters in the word DRAVID such
that the letter R comes before the Ds?

A 720 B 360

C 180 D 120

Answer
02

The total number of ways in which you can arrange the letters in the word
DRAVID is 6!/2! = 360 ways.
Consider the letter R, D, and D.
They can be arranged in three sequences – RDD, DRD, and DDR.
The 360 arrangements will get evenly distributed among the three
sequences.
Among the three sequences, only 1 sequence has R preceding the Ds.
Thus, the number of arraignments in which R comes before the Ds is
360/3 = 120.

Hints
03

In how many ways the word CATHOLICS are arranged such that all that
all the vowels should be in odd positions? (Use nCr and factorial to denote
your answer.)

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03

In how many ways the word CATHOLICS are arranged such that all that
all the vowels should be in odd positions?

5C x 3! x 6!/2!
3

Answer
03

Among the letters in the word CATHOLICS, there are 3 vowels.

To find the arrangements, let’s create 9 placeholders:
_________
oeoeoeoeo
Now, we have 5 placeholders in odd positions.
Since we need only 3, the number of ways in which we can select the
placeholders for our vowels is 5C3 = 10.
For each of the 10 selections, there are 3! ways of arranging the vowels.
Next, we have 6 consonants in which two are of the same type. These
consonants can go to the remaining 6 places. This can be done in:
6!/2! = 360 ways. Thus, the final answer is 5C3 x 3! x 6!/2!

Hints
04

In how many ways can six gem stones be set on a necklace that has a
diamond shaped pendant at the center.

A 6! B 5!

C 6!/2 D 5!/2

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04

In how many ways can six gem stones be set on a necklace that has a
diamond shaped pendant at the center.

A 6! B 5!

C 6!/2 D 5!/2

Answer
04

Normally, for problems based on circular arrangements require you to use

the formula (n – 1)!
However, in this problem, the number of circular arrangements is n!
This is because there is a reference point, the pendant, on the necklace.
Also, there is no distinction between clockwise and anti-clockwise
arrangement because the necklace can be flipped and worn.
Thus, the total number of arrangements is n!/2 = 6!/2.

Hints
05

In how many ways can the letters in the word LETTERS be arranged
such that two Ts are together?

A B

C D

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05

In how many ways can the letters in the word LETTERS be arranged
such that two Ts are together?

A B

C D

Answer
05

To solve this problem, let’s group the two Ts in to a single letter, say x.
Then, we have LEXERS.
The letters in this word can be arranged in 6!/2! ways.
Now, within the group, since the two letters are the same, there’s no more
arrangements possible.
Therefore, the number of arrangements of the letters in the word
LETTERS in which the two Ts are together is 6!/2!

Hints
06

In how many ways 7 different people can sit in 4 buses if any people can
sit any bus ?

A 7^4 B 4^7

C 7C D 7P
4 4

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06

In how many ways 7 different people can sit in 4 buses if any people can
sit any bus ?

A 7^4 B 4^7

C 7C D 7P
4 4

Answer
06

If we ask all the 7 persons to go and sit in the buses one by one, every
person will have 4 choices. He/she can sit in any one of the 4 buses.
So, the number of ways in which one person can sit in which one person
can sit in 4 buses = 4.
The same goes for remaining 6 persons.
So, the total number of ways in which 7 persons can sit in 4 buses = 4
(multiplied 7 times) i.e. 4^7.
So, the answer is 4^7.

Hints
07

In how many ways do 4 boys and 4 girls be arranged in a row such that
no two boys and no two girls sit together?

A 4! x 4! B 2 x 4! x 4!

C 2 x 4! D 5C x 4!
4

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07

In how many ways do 4 boys and 4 girls be arranged in a row such that
no two boys and no two girls sit together?

A 4! x 4! B 2 x 4! x 4!

C 2 x 4! D 5C x 4!
4

Answer
07

There are two cases – (a) The line starts with a boy, and (b) The line
starts with a girl.
CASE A
B_B_B_B_
Here, the letter B marks the places that the boys occupy. Between these
4 places, there are 4! arrangements. Now, the girls occupy the remaining
4 places, and there are 4! ways arranging them. Thus, the total number of
arrangements is 4! x 4!
CASE B
G_G_G_G_
This is a replication of the first case. Therefore, the total number of
arrangements is 4! x 4!. Final answer is 2 x 4! x 4!.

Hints
08

In how many ways can the numbers be printed on a cube such that the
numbers 1 and 6 are on the opposite faces?

A 1 B 2

C 3 D 4

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08

In how many ways can the numbers be printed on a cube such that the
numbers 1 and 6 are on the opposite faces?

A 1 B 2

C 3 D 4

Answer
08

A cube has 6 faces, in which there are 3 pairs of opposite faces. We can
print the numbers 1 and 6 on any of the pairs. Therefore, it might appear
that there are 3 ways. But this is incorrect.
The 3 ways appear different; but they are the same – you can obtain the
new arrangements by rotating the cube. Thus, there is only 1 way of
printing the numbers 1 and 6 on the opposite faces of a cube.

Hints
09

If all the arrangements of the letters in the word MEAT are ordered
alphabetically, what is the rank of the word TEAM?

A 18 B 20

C 21 D 22

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09

If all the arrangements of the letters in the word MEAT are ordered
alphabetically, what is the rank of the word TEAM?

A 18 B 20

C 21 D 22

Answer
09

The total number of arrangements of the letters in the word MEAT is 4!

Among the 24 arrangements, 6 will start with M, 6 with E, 6 with A, and 6
with T.
If we consider the word TEAM, all the words that start with A, E, and M
will come before the words that start with T. Thus, the rank of TEAM is
more than 18.
Now, lets look at the words that start with T.
Keeping T in the first place, the other letters can be arranged in 3! ways.
Among these 6 arrangements, 2 will start with E, 2 with A, and 2 with M.
The words starting with TA will come before TEAM. So, the rank of team
is more than 20.
From the two remaining letters, we know TEAM comes before TEMA.
Thus, the rank is 21.
Hints
10

A person pulls one card from a deck of cards after pulling another one. If
the first two cards are of the same colour, he stops. If they are of different
colours, he pulls two more cards. He continues the process until he stops.
If he stopped after pulling 4 cards, what are the possible number of ways
in which he pulled out the cards?

A 2 B 4

C 6 D 8

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10

A person pulls one card from a deck of cards after pulling another one. If
the first two cards are of the same colour, he stops. If they are of different
colours, he pulls two more cards. He continues the process until he stops.
If he stopped after pulling 4 cards, what are the possible number of ways
in which he pulled out the cards?

A 2 B 4

C 6 D 8

Answer
10

Since he did not stop after pulling the first set of two cards, it means the
cards were of different colours. Thus, we have two cases – RB or BR.
After pulling the second set of two cards, he stops. This means that two
cards were of the same colour. Thus, we have two cases – RR or BB.
Thus, the total number of ways in which he could have pulled the cards is
4.

Hints