Math Is Fun Forum

TARAJS

Member

Registered: 2011-12-01

Posts: 19

Horse Race

Three horses run a race. How many different ways can the three finish if ties are allowed?

I used permutations w/ repetition and got n^r n=3 r=3. so there are 27 different ways the three horses can finish... just want to make sure this is correct.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Horse Race

hi TARAJS,

No, too many.

Permutations would mean (call horses A, B and C)
AAA, AAB, AAC, ABA, ABB, ...............CCC

But a horse can only occupy one position!

If ties weren't allowed:

First place can be any of A, B, C so 3 ways.

Now there are only two choices for 2nd place and no choice about 3rd place.

=> 3 x 2 x 1 = 6 ways altogether.  That's small enough for me to list them.

ABC, ACB, BAC, BCA, CAB, CBA.

But with ties there'll be more.

I'll write a tie like this:  A & B  with C last; or C first followed by A&B.

So here are those possibilities:

A&B C, C A&B, A&C B, B A&C, B&C A, A B&C and finally, A&B&C.

So I get 6 + 6 + 1 = 13 ways.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

TARAJS

Member

Registered: 2011-12-01

Posts: 19

Re: Horse Race

Okay thank you. My only other thing is, what if there are too many to write all out, is there an equation or anything that can work to tell the total or maybe a combination of two equations?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Horse Race

Hi TARAJS;

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Horse Race

hi bobbym

That's what's great fun about this site.  I didn't know that.  But I'm a bit confused by the words between the math:

There is no known formula

closely followed by

they are generated by the formula

I know you like to do at least one impossible thing everyday but writing the formula straight after saying it doesn't exist .....

me =

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Horse Race

Hi Bob;

I hate to confuse people with jargon so I run out of phrases. No closed form like 3n+5 is what I mean. I am not totally sure but a series is not considered a closed form. I would have to explain that the Stirling numbers of the second kind which are in the sum also do not have a formula. Formula is a word people can understand, closed form , elementary form are too much.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.