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Fundamental Counting Principle (Theorem) of Addition

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Counting Principle of Addition

If a total event can be accomplished in two or more mutually exclusive alternative-events (alternative ways), then the number of ways in which the total event can be accomplished is given by the sum of the number of ways in which each alternative-event can be accomplished.

Number of ways in which the total event can be accomplished

= (Number of ways in which the first alternative-event can be accomplished)

+ (Number of ways in which the second alternative-event can be accomplished)

+ (Number of ways in which the third alternative-event can be accomplished)

+ ....

⇒ nE= nEa × nEb × nEc + ....

Eg: 1. Consider choosing a committee of 4 members from a group of 6 men and 5 women such that there are atleast 2 women in the committee.

In finding the number of ways in which the committee can be chosen, we identify that the total event can be accomplished in three alternative ways

  • Total Event (E) = Choosing the committee of 4 members
  • 1st alternative-event (Ea) = Choosing the committee with 2 women and 2 men
  • 2nd alternative-event (Eb) = Choosing the committee with 3 women and 1 man
  • 3rd alternative-event (Ec) = Choosing the committee with 4 women and 0 men

[The events are mutually exclusive because, if the committee is chosen in one of these ways we can say that it was not chosen in the other ways (occurrance of one of these evnets prevents the occurance of the others)]
Choosing the Committee with 4 members
Ea » Choosing the Committee with 2 Women and 2 Men
Eb » Choosing the Committee with 3 Women and 1 Man
Ec » Choosing the Committee with 4 Women and 0 Men

There are three alternative ways for accomplishment of this event

⇒ The number of ways in which the committee can be chosen to have

  • 2 women and 2 men ⇒ nEa = 6C2 × 5C2
    Women × Men
    Available 6 5
    To Choose 2 2
    Choices 6C2 5C2
    These calculations are based on the fundamental counting theorem of multiplication
  • 3 women and 1 men ⇒ nEb = 6C3 × 5C1
    Women × Men
    Available 6 5
    To Choose 3 1
    Choices 6C3 5C1
    These calculations are based on the fundamental counting theorem of multiplication
  • 4 women and 0 men ⇒ nEc = 6C4 × 5C0
    Women × Men
    Available 6 5
    To Choose 4 0
    Choices 6C4 5C0
    These calculations are based on the fundamental counting theorem of multiplication

Therefore,
The number of ways in which the committee can be chosen

= [No. of ways in which the committee with 2 women and 2 men
      (1st alternative event) can be chosen]

+ [No. of ways in which the committee with 3 women and 1 man
      (2nd alternative-event) can be chosen]

+ [No. of ways in which the committee with 4 women and 0 men
      (3rd alternative-event) can be chosen]

⇒ nE = nEa + nEb + nEc
= 6C2 × 5C2 + 6C3 × 5C1 + 6C4 × 5C0
=
6 × 5
2 × 1
 ×
5 × 4
2 × 1
 +
6 × 5 × 4
3 × 2 × 1
 ×
5
1
 +
6 × 5 × 4 × 3
4 × 3 × 2 × 1
 × 1
= 150 + 100 + 15
= 265

Author Credit : The Edifier ... Continued Page 4

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