total setups count/main.tex

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\title{Total Setups Count}
\author{algorembrant}
\date{\today}

\begin{document}
\maketitle

Formula:
\[
\mathcal{S} = \prod_{n=1}^{T} I_{p_n}, \quad S = \{p_1,p_2,\ldots,p_n\}
\]
where:
\begin{align*}
    \mathcal{S} &= \text{the total number of setups in regards to changing its parameter's inputs}\\
    S &= \text{ is a set of all parameters}  \\
    I_{p_n} &= \text{the number of possible input values for parameter } p_n, \\
    p_n &= \text{the } n\text{-th parameter}, \\
    T &= \text{the total number of parameters}.
\end{align*}

Given the following parameters:
\begin{verbatim}
# parameter_n = allPossibleInputs

parameter_1 = 1440   # first parameter and its input
parameter_2 = 60     # second parameter and its input
parameter_3 = 1440   # third parameter and its input
parameter_4 = 1440   # forth parameter and its input
parameter_5 = 60     # fith parameter and its input
parameter_6 = 20     # sixth parameter and its input
\end{verbatim}

We have \( T = 6 \). The total number of setups \( S \) is the product of all possible input values:
\begin{align*}
    S &= I_{p_1} \times I_{p_2} \times I_{p_3} \times I_{p_4} \times I_{p_5} \times I_{p_6} \\
      &= 1440 \times 60 \times 1440 \times 1440 \times 60 \times 20 \\
      &= 1440^3 \times 60^2 \times 20 \\
      &= 214,990,848,000,000 \\
      &\approx 2.1499E+14
\end{align*}


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