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Chapter 33: Problem 1073

Prove this identity: \(\mathrm{P}(\mathrm{n}, \mathrm{n}-1)=\mathrm{P}(\mathrm{n}, \mathrm{n})\).

Short Answer

Expert verified
To prove the identity \(P(n, n-1) = P(n, n)\), we found that the left-hand side simplifies to \(P(n, n-1) = \dfrac{n!}{(n-n+1)!} = n!\) and the right-hand side simplifies to \(P(n, n) = \dfrac{n!}{0!} = n!\). As both sides are equal to \(n!\), the identity is proven: \(P(n, n-1) = P(n, n)\).

Step by step solution

01

Recall the formula for permutations

The formula for a permutation of n objects taken r objects at a time is given by: \(P(n, r) = \dfrac{n!}{(n-r)!}\), where n! represents the factorial of n, which is the product of all positive integers less than or equal to n.
02

Write down the left-hand side of the given identity

We first write down the left-hand side of the given identity, which is \(P(n, n-1)\), and substitute the formula for a permutation: \(P(n, n-1) = \dfrac{n!}{(n-(n-1))!}\).
03

Simplify the left-hand side

Now we simplify the left-hand side: \(P(n, n-1) = \dfrac{n!}{(n-n+1)!} = \dfrac{n!}{1!} = n!\).
04

Write down the right-hand side of the given identity

Next, we write down the right-hand side of the given identity, which is \(P(n, n)\), and substitute the formula for a permutation: \(P(n, n) = \dfrac{n!}{(n-n)!}\).
05

Simplify the right-hand side

Now we simplify the right-hand side: \(P(n, n) = \dfrac{n!}{0!}\). Recall that 0! is defined to be equal to 1, therefore, we have: \(P(n, n) = n!\).
06

Compare both sides of the identity

We have found that the left-hand side of the given identity simplifies to n! (\(P(n, n-1) = n!\)) and the right-hand side also simplifies to n! (\(P(n, n) = n!\)). Thus, both sides of the identity are equal, which proves the given identity: \(P(n, n-1) = P(n, n)\).

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