Compute the following:

a.$\frac{\mathbf{\text{7!}}}{\mathbf{\text{3!}}\mathbf{(}\mathbf{\text{7}}\mathbf{-}\mathbf{\text{3}}\mathbf{)}\mathbf{\text{!}}}$

b.$\left(\begin{array}{l}\text{9}\\ \text{4}\end{array}\right)$

c. $\left(\begin{array}{l}\text{5}\\ \text{0}\end{array}\right)$

d.$\left(\begin{array}{l}\text{4}\\ \text{4}\end{array}\right)$

e.$\left(\begin{array}{l}\text{5}\\ \text{4}\end{array}\right)$

In Mathematics or Statistics, x can be any whole number, and whenever x! is observed, it mainly becomes equal to as follows:

$\mathbf{\text{x!=}}\mathbf{\left(}\mathbf{\text{x}}\mathbf{\right)}\mathbf{\text{×}}\mathbf{(}\mathbf{\text{x}}\mathbf{-}\mathbf{\text{1}}\mathbf{)}\mathbf{\text{×}}\mathbf{(}\mathbf{\text{x}}\mathbf{-}\mathbf{\text{2}}\mathbf{)}\mathbf{\text{×}}\mathbf{...}\mathbf{\text{×}}\mathbf{\left(}\mathbf{\text{2}}\mathbf{\right)}\mathbf{\text{×}}\mathbf{\left(}\mathbf{\text{1}}\mathbf{\right)}$

The value of $\frac{\mathbf{\text{7!}}}{\mathbf{\text{3!}}\mathbf{(}\mathbf{\text{7}}\mathbf{-}\mathbf{\text{3}}\mathbf{)}\mathbf{\text{!}}}$is shown below:

role="math" localid="1653654745275" $$\begin{gathered} \frac{\mathbf{\text{7!}}}{\mathbf{\text{3!}}\mathbf{(}\mathbf{\text{7}}\mathbf{-}\mathbf{\text{3}}\mathbf{)}\mathbf{\text{!}}}\mathbf{\text{=}}\frac{\mathbf{\text{7×6×5×4×3×2×1}}}{\mathbf{\text{3×2×1}}\mathbf{\left(}\mathbf{\text{4}}\mathbf{\right)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{7×6×5×4×3×2×1}}}{\mathbf{\text{3×2×1}}\mathbf{\left(}\mathbf{\text{4×3×2×1}}\mathbf{\right)}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{5040}}}{\mathbf{\text{6×24}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{5040}}}{\mathbf{\text{144}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{5} \end{gathered}$$

Therefore, the value of $\frac{\mathbf{\text{7!}}}{\mathbf{\text{3!}}\mathbf{(}\mathbf{\text{7}}\mathbf{-}\mathbf{\text{3}}\mathbf{)}\mathbf{\text{!}}}$is 3.5.

In Mathematics or statistics , it is often written as for simplicity for calculating. Here, 9! and 4! refer to 9 factorial and 4 factorials, respectively, which are accordingly calculated.

The value of $\left(\begin{array}{l}\text{9}\\ \text{4}\end{array}\right)$is shown below:

role="math" localid="1653655416623" $$\begin{gathered} \mathbf{\left(}\begin{array}{l}\mathbf{\text{9}}\\ \mathbf{\text{4}}\end{array}\mathbf{\right)}\mathbf{\text{=}}\frac{\mathbf{\text{9!}}}{\mathbf{\text{4!}}\mathbf{(}\mathbf{\text{9}}\mathbf{-}\mathbf{\text{4}}\mathbf{)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{9×8×7×6×5×4×3×2×1}}}{\mathbf{\text{4×3×2×1}}\mathbf{\left(}\mathbf{\text{5}}\mathbf{\right)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{9×8×7×6×5×4×3×2×1}}}{\mathbf{\text{4×3×2×1}}\mathbf{\left(}\mathbf{\text{5×4×3×2×1}}\mathbf{\right)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{362,880}}}{\mathbf{\text{24×120}}} \\ \begin{array}{c}\mathbf{\text{=}}\frac{\mathbf{\text{362880}}}{\mathbf{\text{2880}}}\\ \mathbf{\text{=126}}\end{array} \end{gathered}$$

Therefore, the value of $\left(\begin{array}{l}\text{9}\\ \text{4}\end{array}\right)$is 126.

For calculation of$\left(\begin{array}{l}\text{5}\\ \text{0}\end{array}\right)$ , it is sometimes written as$\frac{\mathbf{\text{5!}}}{\mathbf{\text{0!}}\mathbf{\left(}\mathbf{\text{5-0}}\mathbf{\right)}\mathbf{\text{!}}}$ whenever someone tries to calculate. Here, 5! and 0! refer to 5 factorial and 0 factorial, respectively, where 0! is equal to 1.

The calculation of $\left(\begin{array}{l}\text{5}\\ \text{0}\end{array}\right)$is shown below:

$$\begin{gathered} \mathbf{\left(}\begin{array}{l}\mathbf{\text{5}}\\ \mathbf{\text{0}}\end{array}\mathbf{\right)}\mathbf{\text{=}}\frac{\mathbf{\text{5!}}}{\mathbf{\text{0!}}\mathbf{(}\mathbf{\text{5}}\mathbf{-}\mathbf{\text{0}}\mathbf{)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{5×4×3×2×1}}}{\mathbf{\text{1}}\mathbf{\left(}\mathbf{\text{5}}\mathbf{\right)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{5×4×3×2×1}}}{\mathbf{\text{1}}\mathbf{\left(}\mathbf{\text{5×4×3×2×1}}\mathbf{\right)}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{120}}}{\mathbf{\text{120}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=1}} \end{gathered}$$

Therefore, the value of $\left(\begin{array}{l}\text{5}\\ \text{0}\end{array}\right)$is 1.

$\left(\begin{array}{l}\text{4}\\ \text{4}\end{array}\right)$, is often expressed by a person as$\frac{\mathbf{\text{4!}}}{\mathbf{\text{4!}}\mathbf{(}\mathbf{\text{4}}\mathbf{-}\mathbf{\text{4}}\mathbf{)}\mathbf{\text{!}}}$ for doing Mathematical calculations. Here, 4! refers to 4 factorials for both cases and can be calculated mathematically by a person.

The value of $\left(\begin{array}{l}\text{4}\\ \text{4}\end{array}\right)$is shown below:

$$\begin{gathered} \mathbf{\left(}\begin{array}{l}\mathbf{4}\\ \mathbf{4}\end{array}\mathbf{\right)}\mathbf{=}\frac{\mathbf{4}\mathbf{!}}{\mathbf{4}\mathbf{!}\mathbf{(}\mathbf{4}\mathbf{-}\mathbf{4}\mathbf{)}\mathbf{!}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}\mathbf{\times 3\times 2\times 1}}{\mathbf{4}\mathbf{\times 3\times 2\times 1}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{!}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}\mathbf{\times 3\times 2\times 1}}{\mathbf{4}\mathbf{\times 3\times 2\times 1}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{!}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{24}}{\mathbf{24}\mathbf{\times 1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{24}}{\mathbf{24}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1} \end{gathered}$$

Therefore, the calculated value $\left(\begin{array}{l}\text{4}\\ \text{4}\end{array}\right)$is 1.

Anyone can express $\left(\begin{array}{l}\text{5}\\ \text{4}\end{array}\right)$as $\frac{\mathbf{\text{5!}}}{\mathbf{\text{4!}}\mathbf{(}\mathbf{\text{5}}\mathbf{-}\mathbf{\text{4}}\mathbf{)}\mathbf{\text{!}}}$to calculate the final value. Here, 5! and 4! are short forms of 5 factorial and 4 factorials, respectively, used for calculation purposes.

The value of $\left(\begin{array}{l}\text{5}\\ \text{4}\end{array}\right)$is shown below:

role="math" localid="1653657415643" $$\begin{gathered} \left(\begin{array}{l}\text{5}\\ \text{4}\end{array}\right)\mathbf{\text{=}}\frac{\mathbf{\text{5!}}}{\mathbf{\text{4!}}\mathbf{(}\mathbf{\text{5}}\mathbf{-}\mathbf{\text{4}}\mathbf{)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{5×4×3×2×1}}}{\mathbf{\text{4×3×2×1}}\mathbf{\left(}\mathbf{\text{1}}\mathbf{\right)}\mathbf{\text{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=}}\frac{\mathbf{\text{120}}}{\mathbf{\text{24×1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=5}} \end{gathered}$$

Therefore, the actual value of $\left(\begin{array}{l}\text{5}\\ \text{4}\end{array}\right)$is 5.