Question

in combination problems why are we adding the total number of combinations if 'or' is given in the question and multiplying if 'and' is given.explain please.

Answer 1

Fundamental Principles of Counting

Here we shall discuss two fundamental principles viz. principle of addition and principle of multiplication.
These two principles will enable us to understand Permutations and Combinations. In fact these two principles form the base of Permutations and Combinations.


Fundamental Principle of Multiplication

"If there are two jobs such that one of them can be completed in ‘m’ ways, and another one in ‘n’ ways then the two jobs in succession can be done in ‘m X n’ ways."

Example :- In her class of 10 girls and 8 boys, the teacher has to select 1 girl AND 1 boy. In how many ways can she make her selection?
Here the teacher has to choose the pair of a girl AND a boy

For selecting a boy she has 8 options/ways AND that for a girl 10 options/ways
For 1st boy ------- any one of the 10 girls ----------- 10 ways
For 2nd boy ------- any one of the 10 girls ----------- 10 ways
For 3rd boy ------- any one of the 10 girls ----------- 10 ways
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For 8th boy ------- any one of the 10 girls ----------- 10 ways
Total number of ways 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 8b0 ways OR 10 X 8 = 80 ways.

Remark :- The above principle can be extended for any finite number of jobs.


Fundamental Principle of Addition

"If there are two jobs such that they can be performed independently in ‘m’ and ‘n’ ways respectively, then either of the two jobs can be performed in (m + n) ways."

Example :- In her class of 10 girls and 8 boys, the teacher has to select either a girl OR a boy. In how many ways can she make her selection?
Here the teacher has to choose either a girl OR a boy (Only 1 student)

For selecting a boy she has 8 options/ways OR that for a girl 10 options/ways. The first of these can be performed in 8 ways and the second in 10 ways.
Therefore, by fundamental principle of addition either of the two jobs can be performed in (8 + 10) ways. Hence the teacher can make the selection of a student in 18 ways.



Examples 1 :- There are 3 candidates for a classical, 5 for a mathematical, and 4 for a natural science scholarship.

I. In How many ways can these scholarships be awarded?

Clearly classical scholarship can be awarded to anyone of the 3 candidates, similarly mathematical and natural science scholarship can be awarded in 5 and 4 ways respectively. So,
Number of ways of awarding three scholarshipsV= 3 X 5 X 4 = 60 -----------------------[ By Fundamental Principle of Multiplication]

II. In How many ways one of these scholarships be awarded?

Number of ways of awarding one of the three scholarships = 3 + 5 + 4 = 12------------------------[ By Fundamental Principle of Addition]


Example 2 :- A room has 6 doors. In how many ways can a person enter the room through one door and come out through a different
Number of ways coming in the room = 6
Number of ways going out of the room = 5 (He/She cannot go from the same door)

By Fundamental Principle of Multiplication--------> Coming in X Going out = 6 X 5 = 30.