User:IssaRice/Summary of counting techniques: Difference between revisions

Revision as of 02:15, 14 August 2019

Let $A$ be a set with $n$ elements, and let $B$ be a set with $m$ elements.

Description	Set representing counting problem	number of ways to count
	$\{(a_{1},\ldots ,a_{k}):a_{1},\ldots ,a_{k}\in A\}$	$n^{k}$
	$\{\{a_{1},\ldots ,a_{k}\}:a_{1},\ldots ,a_{k}\in A\}$
	$\{(a_{1},\ldots ,a_{k}):a_{1},\ldots ,a_{k}\in A{\text{ and all }}a_{i}{\text{ distinct}}\}$	$P(n,k)={\frac {n!}{(n-k)!}}=n(n-1)\cdots (n-(k+1))$
	$\{\{a_{1},\ldots ,a_{k}\}:a_{1},\ldots ,a_{k}\in A{\text{ and all }}a_{i}{\text{ distinct}}\}$	${\binom {n}{k}}=P(n,k)/(k!)={\frac {n!}{k!(n-k)!}}$
	$\{(a,b):a\in A{\text{ and }}b\in B\}$

@@ Line 4: / Line 4: @@
 ! Description !! Set representing counting problem !! number of ways to count
 |-
-| || <math>\{(a_1, \ldots, a_k) : a_1,\ldots, a_k \in A\}</math> ||
+| || <math>\{(a_1, \ldots, a_k) : a_1,\ldots, a_k \in A\}</math> || <math>n^k</math>
 |-
 | || <math>\{\{a_1, \ldots, a_k\} : a_1,\ldots, a_k \in A\}</math> ||
 |-
-| || <math>\{(a_1, \ldots, a_k) : a_1,\ldots, a_k \in A \text{ and all }a_i\text{ distinct}\}</math> ||
+| || <math>\{(a_1, \ldots, a_k) : a_1,\ldots, a_k \in A \text{ and all }a_i\text{ distinct}\}</math> || <math>P(n,k) = \frac{n!}{(n-k)!} = n(n-1)\cdots (n-(k+1))</math>
 |-
-| || <math>\{\{a_1, \ldots, a_k\} : a_1,\ldots, a_k \in A \text{ and all }a_i\text{ distinct}\}</math>
+| || <math>\{\{a_1, \ldots, a_k\} : a_1,\ldots, a_k \in A \text{ and all }a_i\text{ distinct}\}</math> || <math>\binom n k = P(n,k)/(k!) = \frac{n!}{k!(n-k)!}</math>
 |-
 | || <math>\{(a,b) : a \in A \text{ and } b \in B\}</math> ||