Combinatorics

The study of countable structures.

Using a 4 digit pin gives us 10^4 possible combinations. The chance to guess the correct combination is \frac{1}{10^4} and the average attack would need around \frac{1}{10^4}\div 2 attempts.

Formula-Matrix

	ordered	unordered
put back	n^k	\frac{(k+n-1)!}{k!(n-1)!} = {{k+n-1} \choose k}
keep	\frac{n!}{(n-k)!}	\frac{n!}{(n-k)!\cdot k!}={n\choose k}

Drawing ones

Going with the example of drawing a random number N. Drawing the first number would be a Laplace experiment.

P(N_{i})=\frac{1}{|\ohm|}=\frac{1}{N}

Drawing twice and returning while respecting the order

Every element from a tuple A_{i_j} can represent N states:

|\ohm|=N^n

The probability of any given event happening is:

P(A_{j_{i}},\dots,A_{j_{i}})=\frac{1}{|\ohm|} = \frac{1}{N^n}

Given a set of values [0,1], with one being present 3\times and zero 7\times:

P((1,0,0,0)) = \frac{3\cdot 7^3}{10^4}