How many different ways can a 1st 2nd and 3rd place winner finish a race if there are 10 runners?

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Backing a horse to win or place is as simple - and for many, as good! - as punting gets. Of all the bet types available these are the two most simple to place; they are also the easiest to follow and enjoy as a punter. All you need to do is keep an eye on your selection in running and hope they manage to win or place, in keeping with your bet.

Punters can also back a horse to Win AND Place, this is better known as an Each Way bet.

A Win bet is a bet type that requires the punter to select the horse they believe will be first past the post. If they are correct they will win a sum of money relevant to their investment and the betting product (Fixed Odds or Tote betting). A Place bet is a bet type that requires you to select a horse that finishes in the placings, meaning either 1st, 2nd or 3rd*. * The number of runners in the field will determine the amount of placings that will be paid out. - For fields of seven or fewer runners, the place dividend is only paid out across 1st and 2nd.- For fields of four or fewer runners, there will be no place dividend paid out.If you have already put on a Place bet and then a scratching reduces the field size to seven or fewer, your bet will be subject to the rules applying to the reduced field size. If you have a Place bet on a race with five runners and a scratching(s) reduces the field to four (or fewer), your bet will then be refunded, subject to the rules of each bookmaker and the betting product (Tote or Fixed Odds) selected. The cost of a Win / Place bet is $1 for one full unit, i.e. 100% of the relevant dividend. A punter can take as many units as they like, subject to the gambling agency being prepared to take on liability for the bet. The more units a punter takes, the more the bet will cost them. So for five units a punter would be required to outlay $5. If you place a $5 Win bet on a horse you will receive 500% of whatever betting product - i.e. Tote or Fixed Odds - selected for the wager. Some wagering operators have minimum bets of $1 while others will allow even smaller wagers. Consult individual operators' wagering rules and restrictions for more on this. When wagering on a Win or Place bet, punters may have the choice of Fixed Odds or Tote Betting (including any bookmaker's individual tote product). After selecting your horse, choosing between the tote price and fixed odds is probably the next hardest decision a punter will have to make. The tote price is determined by a betting pool containing the spread of individual bets placed on a particular event. The horse that has attracted the most amount of money will be paying the least, while the horse that has attracted the least amount of money will be paying the highest. The actual tote dividend is only revealed after the race and can often change quite significantly in the lead-up to the race. The closer they are to jumping, the more likely the tote price will reflect the eventual dividend. Fixed odds allow punters to secure a set price on a runner in a given event. Fixed odds can help astute punters lock in value on a runner they suspect is over the odds and is likely to be the subject of significant support among other punters by the close of betting. There is no prescriptive guide or right answer to the question of Tote betting vs fixed odds: the success of either generally depends on your astuteness - and luck - as a punter. Generally speaking though Fixed Odds are at their best when markets first open. And if you get involved with online betting you can compare bookmakers to find the most competitive price out there.
Watching your horse storm home over the top of its rivals - or hold off all challengers in gritty fashion - can be one of life's great joys. Take this gentleman's celebration, for example;

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Permutations and Combinations

Permutations

When you have to take a smaller group of items from a larger group, you often need to know how many different ways there are of making a selection (this comes in handy when studying probability).

In some situations, the order of the items in the smaller group is important. For example, the diagram below shows all possible results for the top three dogs at a dog show:

Each arrangement lists the same dogs, but the order of first-, second-, and third-place winners differ. Arrangements such as these are called permutations. A permutation is an arrangement is which order is important. You can use the counting principle to count permutations.

Counting Permutations

Example 1

You have just downloaded 5 new songs. You can use the counting principle to count the different permutations of those 5 songs. This is the number of different sequences in which you can listen to the new songs on your playlist.

You can listen to the songs in 120 different orders.

Factorials!
In the last example, you evaluated 5 × 4 × 3 × 2 × 1. When you multiply a number by the number 1 less than itself, then by the number 1 less than that, and so on, all the way down to 1, this is known as a factorial. You can write "5 × 4 × 3 × 2 × 1" as "5!" which is read as "5 factorial."
5! = 5 × 4 × 3 × 2 × 1
n! = n × (n − 1) × (n − 2) × ... × 1
Note: The value of 0! is defined as 1.

Example 2

Twelve marching bands entered a competition. In how many different ways can first, second, and third places be awarded?

There are 1,320 different ways to award the three places.

Permutation Notation

The previous example shows how to find the number of permutations of 12 items taken 3 at a time. We can write this as 12P3. In general, the permutation formula is defined as follows:

The number of permutations of n objects taken r at a time = nPr =

Example:

In permutations, the order in which something is arranged is important.

A combination, on the other hand, is a group of items whose order is not important. For example, suppose you go to lunch with a friend. You choose milk, soup, and a salad. Your friend chooses soup, a salad, and milk. The order in which the items are chosen does not matter. You both have same meal.

Listing Combinations

Example

You have 4 tickets to the county fair and can take 3 of your friends. You can choose from Abby (A), Brian (B), Chloe (C), and David (D). How many different choices of groups of friends do you have?

Solution

List all possible arrangements of three friends. Then cross out any duplicate groupings that represent the same group of friends.

You have 4 different choices of groups to take to the fair.

Combination Notation

In Example 1, after you cross out the duplicate groupings, you are left with the number of combinations of 4 items chosen 3 at a time. Using notation, this is written 4C3.

To find the number of combinations of n objects taken r at a time, divide the number of permutations of n objects taken r at a time by r !.

Formula: Example:

Evaluating Combinations

Find the number of combinations if you select 3 items from a group of 8.

8C3

Find the number of combinations if you select 7 items from a group of 9.

9C7

Distinguishing Permutations and Combinations

Examples

State whether the possibilities can be counted using a permutation or combination. Then write an expression for the number of possibilities.

a. There are 8 swimmers in the 400 meter freestyle race. In how many ways can the swimmers finish first, second, and third?

Solution: Because the swimmers can finish first, second, or third, order is important. So the possibilities can be counted by evaluating 8P3.

b. Your track team has 6 runners available for the 4-person relay event. How many different 4-person teams can be chosen?

Solution: Order is not important in choosing the team members, so the possibilities can be counted by evaluating 6C4.

Practice

1. A(n) _?_ is a grouping of objects in which the order is not important.

2. A(n) _?_ is an arrangement of a group of objects in a particular order.

3. There are 12 members of a track team who want to run one of the legs in a 4 person relay race. Choose the calculation that you can use to find the number of ways that runners can be chosen for each of the legs of the relay race.

A.
B.
C.

4. You want to choose 3 different colors of balloons for a party. The balloons are available in 24 colors. How many ways can you choose 3 different colors of balloons?

5. There are 8 students participating in a car wash. How many ways can 2 of the students be chosen to hold signs advertising the car wash?

A. 8 B. 16
C. 28 D. 56

6. A bag contains 1 green marble, 1 blue marble, 1 red marble, and 1 white marble. How many ways can 3 marbles be randomly chosen from the bag, if the order in which the marbles are chosen is important?

Determine whether each situation below describes a permutation or combination. Then find the number of arrangements.

7. Ways to arrange the letters in the word GUITAR.

8. Ways to arrange 7 comic books on a shelf.

9. Ways to choose 4 different fish from 26 kinds of fish.

10. Ways to choose a president, vice-president, treasurer, and secretary from 18 members of a club.

11. Ways to choose 8 students to be extras in a play from 14 students.

12. Ways a coach can arrange the batting order of the 9 starting players of a baseball team

13. A door can be opened only with a security code that consists of five buttons: 1, 2, 3, 4, 5. A code consists of pressing any one button, or any two, or any three, or any four, or all five. How many possible codes are there? (You are to press all the buttons at once, so the order doesn't matter.)

14. How many different 3 digit numbers can you make using the digits 1, 4, 5, 6, 8, and 9, if no digit appears more than once in the number?

Answers

1. combination

2. permutation ("factorial" is also acceptable)

3. A

4. 2024

5. C

6. 24

7. permutation; 720

8. permutation; 5040

9. combination; 14,950

10. permutation; 73,440