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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