SOLUTION: in a race between 7 horses how many ways can the first 3 horses show?

Question 1193831: in a race between 7 horses how many ways can the first 3 horses show?
Found 2 solutions by math_tutor2020, Alan3354:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Answer: 210

Reason:
There are 7 ways to select first place
Then 7-1 = 6 ways to get second place
And 7-2 = 5 ways to get third place

Overall there are 7*6*5 = 210 ways to finish the race

We could use the nPr formula as an alternative path
n = 7 and r = 3
n P r = (n!)/( (n-r)! )
7 P 3 = (7!)/( (7-3)! )
7 P 3 = (7!)/( 4! )
7 P 3 = (7*6*5*4!)/( 4! )
7 P 3 = 7*6*5
7 P 3 = 210
On the second to last line, the "4!" terms canceled and we have the string "7*6*5" show up again.
If you wanted, you can expand out the 4! to get 4*3*2*1, but they cancel out anyway.
Order matters when it comes to the 1st,2nd,3rd place positions. This is why we use nPr instead of nCr.

Some calculators have a specialized nPr button or feature buried inside them somewhere.
On the TI83 for instance, you hit the button labeled "MATH" (unfortunately a very poor choice of naming since everything on a calculator pertains to math) and scroll over to the "PRB" submenu. Then scroll down to nPr. You'll need to type the 7 first before putting the 'nPr' to your home screen. Then type 3 to finish things up and 210 should result.
So you should have 7 nPr 3 on one line and then it results in 210 on the next line.

If you don't have a calculator like a TI83, then you can use a free online one such as this
https://www.calculatorsoup.com/calculators/discretemathematics/permutations.php
I recommend to use it to check your work rather than do everything for you. You won't get practice if you let a machine do all the work.

Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
in a race between 7 horses how many ways can the first 3 horses show?
=================
Show means coming in 3rd
For the 1st 3 horses, it can be one of the 3, unless there's a tie.
So there are 3 ways.
This is true for 7 horses, or for 777 horses.
-------------------------
Chances are, that's not what the problem was meant to be.
Another case of people writing math problems who don't know what they are doing.
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