Question about permutation formula -- How to use the formula when r = 0 ?

AI Thread Summary
The discussion clarifies the application of the permutation formula when r equals 0. It highlights that the formula P^n_r = n(n-1)...(n-r+1) is only valid for r ≥ 1, leading to unreasonable results when r = 0. The correct interpretation shows that P^n_0 equals 1, as it represents the number of ways to arrange n items taken 0 at a time. The empty product convention explains why factorial calculations yield 1 when n is 0. Overall, the key takeaway is that special considerations apply when using permutation formulas for r = 0.
songoku
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Homework Statement
This is not homework

The formula of permutation is ##P^n_r=\frac{n!}{(n-r)!}=n(n-1)...(n-r+1)##

I want to ask how to use the formula when r = 0
Relevant Equations
##P^n_r=\frac{n!}{(n-r)!}=n(n-1)...(n-r+1)##
##P^5_0=\frac{5!}{(5-0)!}=1##

But when I use ##P^n_r=n(n-1)...(n-r+1)##, I get ##P^5_0=5(4)...(6)##

Where is the mistake? Thanks
 
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You use the forlula which is applicable for r ##\ge## 1 and not applicable for r=0. 5>4>3>...>6 is obviously unreasonable.
 
anuttarasammyak said:
You use the forlula which is applicable for r ##\ge## 1 and not applicable for r=0. 5>4>3>...>6 is obviously unreasonable.
Sorry I still don't really understand.

##P^n_r=n(n-1)...(n-r+1)## is derived from ##P^n_r=\frac{n!}{(n-r)!}## and for the later formula there is no restriction ##r \geq 1##

So it means we add a new restriction when deriving the formula ##P^n_r=n(n-1)...(n-r+1)##?

Thanks
 
The formula <br /> n! = \prod_{r=1}^n r for n &gt; 0 gives the familiar <br /> n! = n \times (n-1) \times \dots \times 2 \times 1. But for n = 0 it gives the empty product which by convention is \displaystyle \prod_{r \in \emptyset} r = 1. So it should not surprise you that formulae derived for cases where arguments of ! are strictly positive may not apply when an argument of ! is zero.
 
Thank you very much for the explanation anuttarasammyak and pasmith
 
Another explanation is that ##P^5_0=\frac{5!}{(5-0)!} = \frac{5!}{5!} = 1##
In words, the number of permutations of 5 things taken 0 at a time is, per the above, 1.
 
songoku said:
##n(n-1)...(n-r+1)##
… means: start with 1, then multiply it by n, then by n-1, etc., but don't go below n-r+1.
With r=0, n is already below n-r+1, so stop at 1.
 
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