Counting¶
Fundamental counting principle¶
No of ways \(= n_1 \times n_2 \times \dots\)
How many different \(x\) digit numbers can be created using \(n\) digits only: \(y^x\)
Factorial¶
\(n\) unique objects can be arranged in \(n!\) ways
Restrictions¶
No of ways to follow rule = no of ways to ignore rule - no of ways to break rule
Duplicates¶
When there are duplicate objects $$ \begin{aligned} &= \dfrac{n!}{\prod_i^k n_i!} \end{aligned} $$ where
- \(k=\) no of unique groups
- \(n_i =\) no of objects in group \(k\)
- \(\sum_i^k n_i = n\)
Permutation¶
Combination¶
- Order doesn’t matter
- Outcomes of one stage are same as outcomes of other stages