Dell Permutation & Combination Questions

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.

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

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

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.

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

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

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

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

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

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