I don't think (n + 1) * n! = [n * (n + 1)]!Pual Black said:Homework Statement
Hello
I have to proof this
##(n+1)nPr=(n+1)P(r+1)##
The attempt at a solution
if i substitute this in
##nPr=\frac{n!}{(n-r)!}##
I get this
##(n+1)nPr=\frac{[(n+1)n]!}{[(n+1)n-r]!}##
This will give
##(n+1)nPr=\frac{(n^2+n)!}{(n^2+n-r)!}##
Now i don't know how to continue.
Yes. It's (n+1)(nPr).Pual Black said:Did you mean that (n+1)nPr isn't the same as ((n+1)n)Pr ??
Yes. And since you are dealing with equalities, not inequalities, it doesn't matter which side you start with to arrive at the other.Pual Black said:Ok thanks. Now let's go on
##(n+1)nPr=(n+1)\frac{n!}{(n-r)!}##
I have to get this
##\frac{(n+1)!}{[(n+1)-(r+1)]!}##
to proof the R.H.S
But if i start with the Right side
##(n+1)P(r+1)=\frac{(n+1)!}{[(n+1)-(r+1)]!}=(n+1)\frac{n!}{(n-r)!}=(n+1)nPr=L.H.S##
But how to start with left side and proof the right side ?
I can -1 +1 to the denominator to get the (n+1)-(r+1)
But can i say that (n+1)n!=(n+1)! ?
I think yes of course
Ok. Thank you very much for your help.haruspex said:Yes. And since you are dealing with equalities, not inequalities, it doesn't matter which side you start with to arrive at the other.
A permutation proof is a type of mathematical proof that uses the concept of permutations to show that an equation or statement is true. It involves manipulating the order or arrangement of a set of objects or elements to prove that the equation holds.
To prove this equation using permutation, you would start by expanding both sides using the definition of nPr (n permutation r). Then, you would use the commutative and associative properties of multiplication to rearrange the terms in a way that allows you to cancel out common factors. Finally, you would use the definition of (n+1)P(r+1) to show that the two sides are equal.
The steps involved in a permutation proof are as follows:
Yes, an example of a permutation proof would be proving that nPr = n!/(n-r)! for all values of n and r. This can be done by expanding both sides of the equation, rearranging the terms, and using the definition of factorial to show that the two sides are equal.
Permutation is important in mathematics because it allows us to analyze and manipulate the order or arrangement of a set of objects or elements. This is useful in various areas of mathematics, such as combinatorics, probability, and group theory. Permutation proofs are also commonly used in mathematics to show that equations or statements are true.