Chapter 13, Problem 10CT

To determine

To calculate: The number of ways one could make an exacta bet, in horse racing, an exacta bet requires the better to pick the first two horses in the exact order, if there are $8$ horses in a race.

Answer to Problem 10CT

Solution:

There are $56$ different exacta bets are possible.

Explanation of Solution

Given information:

In horse racing, an exacta bet requires the better to pick the first two horses in the exact order, if there are $8$ horses in a race.

Formula used:

The number of arrangements of $n$ objects using $r\le n$ of them, in which the $n$ objects are distinct, repetition of objects is not allowed and order is important is given by the formula,

$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

In horse racing, an exacta bet requires the better to pick the first two horses in the exact order, if there are $8$ horses in a race.

Since the order of the horses matters use permutation to select two horses in the exact order from $8$ horses.

Thus, here $n=8$ and $r=2$.

Using the formula, $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$.

$P\left(8,2\right)=\frac{8!}{\left(8-2\right)!}$.

$\Rightarrow P\left(8,2\right)=\frac{8!}{6!}$.

By writing $8!$ in factorial notation,

$\Rightarrow P\left(8,2\right)=\frac{8\cdot 7\cdot 6!}{6!}$.

$\Rightarrow P\left(8,2\right)=8\cdot 7=56$.

Therefore, there are $56$ exacta bets possible.