Counting

Last Updated: 10 months ago2024-05-14 ; Contributors: AhmedThahir

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

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