Combination of 'n' different things taken 'r' at a time

COMBINATION

Combination means selection. Let P, Q, R be three letters, and then we can combine any two of them in following ways:

AB, BC, AC

Combination of ‘n’ different things taken ‘r’ at a time

            The number of all combinations of n distinct things taken r at a time (r<= n ) is

nCr = n!/(n-r)!r! ,   C denotes combination

Problems

1.      How many different committees of 6 members may be formed from 5 men and 4 ladies?

Solution:

 Totally there are 9 persons we need to select  any 5 persons.

= 9C6

= 9!/3!x6!

2.      In a test paper there are total 10 questions. In how many different ways can you choose 6 questions to answer?

Solution:

Out of 10 questions, 6 questions can be selected in 10C6 ways

10C6 = 210

3.      Arun has 5 friends. In how many ways can she invite one or more of them to a dinner?

Solution:

She may invite one or more friends be selecting either 1 or 2 or 3 or 4 or 5 friends out of 5 friends.

Therefore 1 friend can be selected out of 5 in 5C1 ways

2 friends can be selected out of  5 in 5C2 ways . similarly its goes on

Hence the required number of ways

= 5C1+ 5C2 + 5C3 + 5C4 + 5C5

= 5+10+10+5+1 =31

4.      An examinee is required to answer six questions out of 12 questions which are divided into 2 groups each containing 6 questions and he is not permitted to answer more the 4 questions from any group. In how many ways can he answer 6 questions?

Solution:

Required number of ways

 = (6C4x6C2) +(6C3x6C3)+(6C2x6C4)

= (15x15)+(20x20)+(15x15)

=225+400+225

=850