Question

Find the value of each of these quantities:

a) P (6,3)

b) P (6,5)

c) P (8,1))

d) P 8,5)

e) P (8,8)

f) P (10,9)

Short answer

(a) The resultant answer is120.

(b) The resultant answer is 720.

(c) The resultant answer is 08.

(d) The resultant answer is 6720.

(e) The resultant answer is 40,320.

(f) The resultant answer is 3,628,800.

Step-by-step solution

Given data

The given data is the quantities.

Concept of Permutation

Definition permutation (order is important):$\mathbf{P}\mathbf{(}\mathbf{n}\mathbf{,}\mathbf{r}\mathbf{)}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}$

with $\mathbf{n}\mathbf{!}\mathbf{=}\mathbf{n}\mathbf{\cdot }\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\cdot }\mathbf{\dots }\mathbf{\cdot }\mathbf{2}\mathbf{\cdot }\mathbf{1}$.

Evaluate the definition of a combination

(a)

Evaluate the definition of a permutation:

$\begin{array}{r}P(6,3)=\frac{6!}{(6-3)!}\\ =\frac{6!}{3!}\\ =120\end{array}$

Evaluate the definition of a combination

(b)

Evaluate the definition of a permutation:

$\begin{array}{r}P(6,5)=\frac{6!}{(6-5)!}\\ =\frac{6!}{1!}\\ =6!\\ =720\end{array}$

Evaluate the definition of a combination

(c)

Evaluate the definition of a permutation:

$\begin{array}{r}P(8,1)=\frac{8!}{(8-1)!}\\ =\frac{8!}{7!}\\ =8\end{array}$

Evaluate the definition of a combination

(d)

Evaluate the definition of a permutation:

$\begin{array}{r}P(8,5)=\frac{8!}{(8-5)!}\\ =\frac{8!}{3!}\\ =6720\end{array}$

Evaluate the definition of a combination

(e)

Evaluate the definition of a permutation:

$\begin{array}{r}P(8,8)=\frac{8!}{(8-8)!}\\ =\frac{8!}{0!}\\ =8!\\ =40320\end{array}$

Evaluate the definition of a combination

(f)

Evaluate the definition of a permutation:

$\begin{array}{r}P(10,9)=\frac{10!}{(10-9)!}\\ =\frac{10!}{1!}\\ =10!\\ =3628800\end{array}$