Shortcut 91:
Events
Dependent=multiplication

Question:
There 3 busses from city A to city B and there are 5 busses from city B to city C. In how many ways a person can travel from city A to C through B?
Choosing a bus from city B depends on choosing a bus from city A.
Number of ways = 3 x 5 = 15
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Question:
There 3 busses from city A to city B and there are 5 busses from city A to city C. In how many ways a person can travel from city A to C or B?
Choosing a bus to city B or C are not dependent.
Number of ways = 3 + 5 = 8

Shortcut 92:
Arrangement with repetition
$$n^n;n^r$$
Question:
In how many ways the letters of the word “ORANGE” can be arranged with repetition?
n = 6   (n = total number of elements)
Since all the elements are taken,
Number of arrangements = 66
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Question:
In how many ways three letters from the word “ORANGE” can be arranged with repetition?
n = 6; r = 3    (r = number of elements taken for arrangement)
Number of arrangements = 63 = 216

Shortcut 93:
Arrangement without repetition
$$n!;nP_r=n!/(n-r)!$$
Question:
In how many ways the letters of the word “MANGO” can be arranged without repetition?
n = 5
Since all the elements are taken for arrangement,
Number of elements = n! = 5! = 120
Question:
In how many ways any three letters of the word ‘MANGO” can be arranged without repetition?
n= 5; r = 3
nPr = 5!/(5 – 3)! = 120/2 = 60 ways

Shortcut 94:
Elements occurring together
2!*(n-1)

3!*(n-2)
Question:
In how many ways letters of the word “ORANGE” arranged so that the vowels will always occur together?