Combinations


 
 
Concept Explanation
 

Combination

Each of the groups or selections that can be made by taking some or all of a number of things without considering the arrangement or order is called a combination.

Thus, the selections that can be made by taking the letters a, b and c two at a time are three. i.e., ab, bc and ac. These three groups or selections are called the combination of three things taken two at a time.

The number of combinations of n dissimilar things taken r at a time is calculated using the expression ^nC_r.

  ^nC_r=frac{n!}{r!times (n-r)!}

For Example: The number of selection of 6 dissimilar things taken 4 at a time is

                         ^6C_4=frac{6!}{4!times 2!}

                                  =frac{6times 5times 4times 3times 2times 1}{4times 3times 2times 1times 2times 1}=15

The number of combinations of n dissimilar things taken all at a time is ^nC_n.

                           ^nC_n=frac{n!}{n!(n-n)!}=frac{n!}{n!0!}=1

Also,  ^nC_0=1; and ;^nC_r=^nC_{n-r}

For Example: The number of selection of 3 dissimilar things taken all at is a time

^3C_3=frac{3!}{3!times0!}=1.

Illustration: Each person's performance compared with all other persons is to be done to rank them subjectively. How many comparisons are needed in total, if there are 11 persons ?

(a) 66          (b) 55          (c) 54        (d) 45

Answer : b

Solution Comparison is always done between two peoples, we have to select 2 persons among the 11 persons for a comparison. Hence

Total ways = ^{11}C_2=frac{11!}{2! times(11-2)!}=frac{11!}{2! times 9!}=frac{11times 10}{2}=55

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Sample Questions
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Question : 1

 How many committee of five persons with a chairperson can be selected form 12 persons?
 

Right Option : B
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