Question

Find the value of each of these quantities.

a) \(C(5,1)\)

b) \(C(5,3)\)

c) \(C(8,4)\)

d) \(C(8,8)\)

e) \(C(8,0)\)

f) \(C(12,6)\)

Short answer

(a) The resultant answer is5.

(b)The resultant answer is10.

(c)The resultant answer is70.

(d)The resultant answer is1.

(e)The resultant answer is1.

(f) The resultant answer is 924.

Step-by-step solution

Given data

The given data is 12 horses.

Concept of Permutation

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

Definition combination (order is not important):$\mathbf{C}\mathbf{(}\mathbf{n}\mathbf{,}\mathbf{r}\mathbf{)}\mathbf{=}\left(\begin{array}{l}n\\ r\end{array}\right)\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}$

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

The order of the horses is important (since we want to determine the first, second and third position), thus we need to use the definition of permutation.

We will select 3 horses from the 12 horses (as we want to determine the top 3).

n = 12

r = 3

Evaluate the definition of a permutation:

$\begin{array}{r}P(12,3)=\frac{12!}{(12-3)!}\\ P(12,3)=\frac{12!}{9!}\\ P(12,3)=12\cdot 11\cdot 10\\ P(12,3)=1320\end{array}$