Fix factorial symbol and change variable k to n

Makefile

@@ -1,16 +1,22 @@
 PRESENTATIONS := $(patsubst src/%/presentation.tex,build/presentation_%.pdf,$(wildcard src/*/presentation.tex))
 HANDOUTS := $(patsubst build/presentation_%.pdf,build/presentation_%_handout.pdf,$(PRESENTATIONS))
 
+RC_PDFLATEX := $(shell grep '$$pdflatex' .latexmkrc \
+	       | sed -e 's/.*"\(.*\)".*/\1/' -e 's/%S//' -e 's/%O//')
+
 .PHONY: all
 all: $(PRESENTATIONS) $(HANDOUTS)
 
 build/presentation_%.pdf: src/%/presentation.tex build/prepared
-	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS latexmk $<
+	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
-	mv build/presentation.pdf $@
+		latexmk -outdir=build/$* $<
+	cp build/$*/presentation.pdf $@
 
 build/presentation_%_handout.pdf: src/%/presentation.tex build/prepared
-	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS latexmk -pdflatex='pdflatex %O "\def\ishandout{1}\input{%S}"' $<
+	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
-	mv build/presentation.pdf $@
+		latexmk -outdir=build/$*_handout \
+		-pdflatex='$(RC_PDFLATEX) %O "\def\ishandout{1}\input{%S}"' $<
+	cp build/$*_handout/presentation.pdf $@
 
 build/prepared:
 	mkdir build

src/2026-01-16/presentation.tex

@@ -30,7 +30,10 @@
 \usepackage{tikz}
 \usepackage{tikz-3dplot}
 \usetikzlibrary{spy, external, intersections, positioning}
+
+\ifdefined\ishandout\else
 \tikzexternalize
+\fi
 
 \usepackage{pgfplots}
 \pgfplotsset{compat=newest}
@@ -139,7 +142,7 @@
     \end{gather*}
     \vspace*{-2mm}
     \begin{gather*}
-        P_X(k) = \frac{\lambda^k}{k!}e^{-\lambda} \\[2mm]
+        P_X(n) = \frac{\lambda^n}{n!}e^{-\lambda} \\[2mm]
         \phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right)
     \end{gather*}
     \vspace*{-2mm}
@@ -160,7 +163,7 @@
             \vspace*{-6mm}
             \begin{gather*}
                 X \sim \text{Poisson}(\lambda) \\[3mm]
-                P_X(k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!} \\[4mm]
+                P_X(n) = \frac{\lambda^n \cdot e^{-\lambda}}{n!} \\[4mm]
                 \phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right)
             \end{gather*}
         \end{greenblock}
@@ -168,7 +171,7 @@
             \vspace*{-6mm}
             \begin{gather*}
                 \nsum_{k=0}^{n} \binom{n}{k}a^k b^{n-k} = (a+b)^n, \hspace{15mm}
-                \binom{n}{k} = \frac{n!}{(n-k!)k!}
+                \binom{n}{k} = \frac{n!}{(n-k)!k!}
             \end{gather*}
         \end{greenblock}
         \column{\kitthreecolumns}
@@ -228,11 +231,11 @@
             zweier Zufallsvariablen.
             \pause\begin{gather*}
                 X \sim \text{Poisson}(\lambda_1) \hspace{3mm}
-                    \Leftrightarrow \hspace{3mm} P_X(k)
+                    \Leftrightarrow \hspace{3mm} P_X(n)
-                    = \frac{\lambda_1^k \cdot e^{-\lambda_1}}{k!} \hspace{30mm}
+                    = \frac{\lambda_1^n \cdot e^{-\lambda_1}}{n!} \hspace{30mm}
                 Y \sim \text{Poisson}(\lambda_2) \hspace{3mm}
-                    \Leftrightarrow \hspace{3mm} P_Y(k)
+                    \Leftrightarrow \hspace{3mm} P_Y(n)
-                    = \frac{\lambda_2^k \cdot e^{-\lambda_2}}{k!}
+                    = \frac{\lambda_2^n \cdot e^{-\lambda_2}}{n!}
             \end{gather*}
             \pause\begin{align*}
                 P_Z(n) &= P_{X+Y}(n) = \nsum_{k=0}^{n} P_X(k)P_Y(n-k)
@@ -416,7 +419,7 @@
                 \begin{lightgrayhighlightbox}
                     Erinnerung: Unabhängige Ereignisse
                     \begin{align*}
-                        X,Y \text{ \normalfont unabhängig}
+                        A,B \text{ \normalfont unabhängig}
                         \hspace{5mm} \Leftrightarrow \hspace{5mm}
                         P(AB) = P(A)P(B)
                     \end{align*}
@@ -436,7 +439,8 @@
                 \begin{lightgrayhighlightbox}
                     Erinnerung: Varianz
                     \begin{align*}
-                        V(X) = E\big( \left(X - E(X)\right)^2 \big) = E(X^2) - E^2(X)
+                        V(X) = E\big( \left(X - E(X)\right)^2 \big) =
+                        E(X^2) - E^2(X)
                     \end{align*}
                     \vspace*{-13mm}
                 \end{lightgrayhighlightbox}