Basics of Counting

If sets A and B are disjoint sets(the sets which have no common element in them), then

|A⋃B|=|A| + |B|

For example:

a class has 43 girls and 54 boys total of the class=43+54=97

If set A and set B the cartesian product is :

A*B={(a,b)| a ∈ A and b ∈ B}

if |A| =n and |B| = m then |A*B|=n*m

for example:

A={a,b} and B={c,d}

A*B={(a,c),(a,d),(b,c),(b,d)}

Examples on counting:

1. how many password satisfy the following requirements:

solution:

first we will find the six character password as:

=52*62*62*62*62*62 { here first 52 is for 26 small letters and 26 for capital letters, other 5 times 62 are for digits(10) and letters(52=26+26)}

=52*62⁵

second for seven characters password as:

=52*62*62*62*62*62*62{ here one extra 62 is multiplied according to the 1st case it is for the seventh characters}

=52*62⁶

last for eight characters password as:

=52*62*62*62*62*62*62*62 { here one extra 62 is multiplied according to the 2nd case it is for the eighth characters}

=52*62⁷

in total the three cases are added to get total number of passwords:

=52*62⁵+52*62⁶+52*62⁷

=19*10¹⁴

2. how many 4 digit numbers with first 5 digits?

solution:

here, for the first place we have all 5 choices and for 2nd also 5 choices and so on...

=5*5*5*5=625

Mathematics