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**Hint:**We have to evaluate ${}^{5}{{P}_{4}}$ which is the permutation and of the form ${}^{n}{{P}_{r}}$ and we know that the expansion of ${}^{n}{{P}_{r}}$ is equal to $\dfrac{n!}{\left( n-r \right)!}$. Now, substitute n as 5 and r as 4 in this formula to get the value of ${}^{5}{{P}_{4}}$. Also, the expansion of $n!$ is equal to $n\left( n-1 \right)\left( n-2 \right).....3.2.1$ so use this expansion to solve the factorials in the formula of ${}^{n}{{P}_{r}}$.

**Complete step-by-step solution:**

We have to evaluate the following:

${}^{5}{{P}_{4}}$

The above expression is the permutation and of the following form:

${}^{n}{{P}_{r}}$

The expansion of the above expression in terms of factorial is as follows:

$\dfrac{n!}{\left( n-r \right)!}$

Substituting n as 5 and r as 4 in the above expression we get,

$\Rightarrow \dfrac{5!}{\left( 5-4 \right)!}$

The above expression is the expansion of ${}^{5}{{P}_{4}}$ so simplifying the above expression we get,

$\Rightarrow \dfrac{5!}{\left( 1 \right)!}$

The expansion of $5!$ is as follows:

$\begin{align}

& =5.4.3.2.1 \\

& =120 \\

\end{align}$

And the value of $1!$ is equal to 1 so substituting the value of $5!\And 1!$ in $\dfrac{5!}{\left( 1 \right)!}$ we get,

$\begin{align}

& =\dfrac{120}{1} \\

& =120 \\

\end{align}$

From the above, we have evaluated ${}^{5}{{P}_{4}}$ as 120.

**Hence, the correct option is (b).**

**Note:**The significance and meaning of the expression ${}^{5}{{P}_{4}}$ written in the above problem is that it means these are the possible ways to arrange 4 persons in 5 chairs. And here, all the 5 chairs are different. So, from this we can learn the concept of arrangement of n persons in r chairs or n persons in n rows.

As for permutations or arrangement of things we use ${}^{n}{{P}_{r}}$ so for combinations or selections we use ${}^{n}{{C}_{r}}$. This expression ${}^{n}{{C}_{r}}$ means the number of possible ways of selecting r items from n items in which order does not matter.

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