From bc1263e509d8fd8338bf93069ea7c342f6751278 Mon Sep 17 00:00:00 2001
From: "Pavly Gerges (pavl_g)" <60224159+pavly-gerges@users.noreply.github.com>
Date: Fri, 17 Jan 2025 15:37:57 +0200
Subject: [PATCH] introduction.md: permutation over repeated a pattern

---
 embedded-system-design/theory-of-computation/introduction.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/embedded-system-design/theory-of-computation/introduction.md b/embedded-system-design/theory-of-computation/introduction.md
index 68f569c..b42d3be 100644
--- a/embedded-system-design/theory-of-computation/introduction.md
+++ b/embedded-system-design/theory-of-computation/introduction.md
@@ -42,7 +42,7 @@ This document will present some prerequisite definitions and formal representati
    * A permutation of A is the number of all the possible subsequences of A taken $$|A|$$ at a time; **given the property of no repetition of the permutated objects**:
       $$_nP_n = \prod\_{i = 0}^{(n - 1)} (n - i) = \frac{n!}{(n - n)!} = n! = n.(n-1).(n-2).(n-3)...(n-(n-2)).(n-(n-1))$$. 
    * A permutation of A taken _r_ at a time is the number of all the possible subsequences of A taken _r_ at a time (aka. subsequences constructed with cardinality of _r_); such that $$1 <= r <= n$$, and **given the property of no repetition of the permutated objects**: $$_rP_n = \prod\_{i = 0}^{(r - 1)} (n - i) = \frac{n!}{(n - r)!} = n.(n-1).(n-2).(n-3)....(n-(r-2)).(n-(r-1))$$.
-   * A permutation of A taken _r_ at a time is the number of all the possible subsequences of A taken _r_ at a time; such that $$1 <= r <= n$$, and **given the property of allowed repetitions of the permutated objects**; such that the first member $m_0$ repeats $k_0$ number of times, the second member $m_1$ repeats $k_1$ number of times, and so on till member $m_n$; where n is the total number of objects in set A: $$
+   * A permutation of A taken _r_ at a time is the number of all the possible subsequences of A taken _r_ at a time; such that $$1 <= r <= n$$, and **given the property of allowed repetitions of the permutated objects**; such that the first member $m_0$ repeats $k_0$ number of times, the second member $m_1$ repeats $k_1$ number of times, and so on till member $m_n$; where n is the total number of objects in set A: $$_rP_n^{\\{k_0, k_1, ...k_n\\}} = \frac{\prod\_{i = 0}^{(r - 1)} (n - i)}{\prod\_{m = 0}^{n} (\prod\_{j = 0}^{(k - 1)} (k - j))_m} = \frac{n!}{k_0.k_1...k_n.(n - r)!}$$.
 
 
 ## Combinatorics