Question #1
Out of 7 consonants and 4 vowels, how many words of 3 consonants
and 2 vowels can be formed?
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 ?
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?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?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?
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?
'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?
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?
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?
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
Required number of ways =
= (8C5*10C6) = (8C3*10C4)
= 11760
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