Permutations and Combinations - 1
The Addition Principle (AP):
One can reach city 'A' from city 'B' by sea, air and road. Suppose that there are 2 ways by sea, 3 ways by air and 2 ways by road. Then by (AP), the total number of ways 'A' to 'B' by sea, air or road is 2+3+2=7.
(AP): Assume that there are
'n₁' ways for the event 'E₁' to occur,
'n₂' ways for the event 'E₂' to occur,
. . . .
. . . .
'nₖ' ways for the event 'Eₖ' to occur,
Where k greater than or equal to 1.
If these ways for the different events to occur are pair wise disjoint, then the number of ways for at least one of the events E₁,E₂,E₃.......or Eₖ to occur is
n₁+n₂+...............+nₖ = ∑ᵢ₌₁ᵏ nᵢ
To reach city 'D' from city 'A', one has to pass through city 'B' and then city 'C'. If there are two ways to travel from 'A' to 'B', 5 ways from 'B' to 'C' and '3' ways from 'C' to 'D', then by (MP), the number of ways from 'A' to 'D' via 'B' to 'C' is given by 2×5×3=30.
(MP) : Assume that an event 'E' can be
decomposed into 'r' ordered events
E₁,E₂,........Eᵣ, and there are,
'n₁' ways for the event 'E₁' to occur,
'n₂' ways for the event 'E₂' to occur,
. . . .
. . . .
'nₖ' ways for the event 'Eᵣ' to occur,
Then the total number of ways for the event 'E' to occur is given by
n₁×n₂×...............×nₖ = Πᵢ₌₁ʳ nᵢ
