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
	${(a_{1}, \dots, a_{k}) : a_{1}, \dots, a_{k} \in A}$	$n^{k}$
	${{a_{1}, \dots, a_{k}} : a_{1}, \dots, a_{k} \in A}$
	${(a_{1}, \dots, a_{k}) : a_{1}, \dots, a_{k} \in A and all a_{i} distinct}$	$P (n, k) = \frac{n!}{(n - k)!} = n (n - 1) \dots (n - (k + 1))$
	${{a_{1}, \dots, a_{k}} : a_{1}, \dots, a_{k} \in A and all a_{i} distinct}$	$(\binom{n}{k}) = P (n, k) / (k!) = \frac{n!}{k! (n - k)!}$
	${(a, b) : a \in A and b \in B}$

@@ 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> ||