Dell Permutation & Combination Questions

Question #1

Out of 7 consonants and 4 vowels, how many words of 3 consonants 
and 2 vowels can be formed?
Question Explanation

Number of ways of selecting (3 consonants out of 7)
and (2 vowels out of 4) = (7C3*4C2)
= 210.
Number of groups, each having
3 consonants and 2 vowels = 210.
Each group contains 5 letters.

Number of ways of arranging 5 letters among
themselves = 5! = 120

Required number of ways = (210 x 120)
= 25200.

Question #2

A committee of 5 persons is to be formed from 6 men and 4 women. 
In how many
ways can this be done when at least 2 women are
included ?
Question Explanation

When at least 2 women are included.
The committee may consist of 3 women, 2 men :
It can be done in 4C3*6C2 ways
or, 4 women, 1 man : It can be done in 4C4*6C1ways
or, 2 women, 3 men : It can be done in 4C2*6C3 ways.
Total number of ways of forming the committees
= 4C2*6C3+4C3*6C2+4C4*6C1
= 6 x 20 + 4 x 15 + 1x 6
= 120 + 60 + 6 =186

Question #3

In how many different ways can the letters of the word 'LEADING' 
be arranged in such a way that the vowels always come together?
Question Explanation

The word 'LEADING' has 7 different letters.
When the vowels EAI are always together,
they can be supposed to form one letter.
Then, we have to arrange the letters LNDG (EAI).
Now, 5 (4 + 1) letters can be arranged in 5! = 120 ways.
The vowels (EAI) can be arranged among
themselves in 3! = 6 ways.
Required number of ways = (120 x 6) = 720

Question #4

n how many different ways can the letters of the word 'OPTICAL' 
be arranged so that the vowels always come together?
Question Explanation

The word 'OPTICAL' contains 7 different letters.
When the vowels OIA are always together,
they can be supposed to form one letter.
Then, we have to arrange the letters PTCL (OIA).
Now, 5 letters can be arranged in 5! = 120 ways.
The vowels (OIA) can be arranged among
themselves in 3! = 6 ways.
Required number of ways = (120 x 6) = 720.

Question #5

A college has 10 basketball players. A 5-member team and a captain 
will be selected out of these 10 players. How many different
selections can be made?
Question Explanation

A team of 6 members has to be
selected from the 10 players.
This can be done in 10C6 or 210 ways.
Now, the captain can be selected from
these 6 players in 6 ways.
Therefore, total ways the selection
can be made is 210×6= 1260

Question #6

How many 4-letter words with or without meaning, can be formed 
out of the letters of the word, 'LOGARITHMS', if repetition
of letters is not allowed?
Question Explanation

'LOGARITHMS' contains 10 different letters.
Required number of words = Number of
arrangements of 10 letters, taking 4 at a time.
= 10P4
= 5040.

Question #7

12 people at a party shake hands once with everyone else in the 
room.How many handshakes took place?
Question Explanation

There are 12 people, so this is our n value.
So, 12C21 = 66

Question #8

In how many ways can a group of 5 men and 2 women be made out 
of a total of 7 men and 3 women?
Question Explanation

Required number of ways =
= (7C5*3C2) = 63

Question #9

There are 7 non-collinear points. How many triangles can be 
drawn by joining these points?
Question Explanation

A triangle is formed by joining any
three non-collinear points in pairs.
There are 7 non-collinear points
The number of triangles formed = 7C3 = 35

Question #10

In how many ways a committee, consisting of 5 men and 6 women 
can be formed from 8 men and 10 women?53400
Question Explanation

Required number of ways =
= (8C5*10C6) = (8C3*10C4)
= 11760

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