\( { } ^ { n } P _ { r } - { } ^ { - } P _ { r - 1 } = ( n - r ) ^ { n } P _ { r - 1 } \)

To prove the equation \[ { } ^{ n } P_{ r } - { } ^{ n } P_{ r - 1 } = ( n - r ) { } ^{ n } P_{ r - 1 } \] we start by recalling the definition of permutations. The number of ways to arrange \( r \) objects from \( n \) distinct objects is given by: \[ { }^{ n } P_{ r } = \frac{n!}{(n - r)!} \] Similarly, the number of ways to arrange \( r - 1 \) objects from \( n \) distinct objects is: \[ { }^{ n } P_{ r - 1 } = \frac{n!}{(n - (r - 1))!} = \frac{n!}{(n - r + 1)!} \] Now, substituting these definitions into the left-hand side of the equation: \[ { }^{ n } P_{ r } - { }^{ n } P_{ r - 1 } = \frac{n!}{(n - r)!} - \frac{n!}{(n - r + 1)!} \] To combine these fractions, we need a common denominator, which is \((n - r)!(n - r + 1)\): \[ = \frac{n! (n - r + 1) - n!}{(n - r)! (n - r + 1)} \] Simplifying the numerator: \[ = \frac{n! (n - r + 1 - 1)}{(n - r)! (n - r + 1)} = \frac{n! (n - r)}{(n - r)! (n - r + 1)} \] Now, we can simplify \( n! \) in the numerator: \[ = \frac{n (n - 1)! (n - r)}{(n - r)! (n - r + 1)} \] Now, we can express \( (n - r)! \) in terms of \( (n - r + 1)! \): \[ = \frac{n (n - 1)! (n - r)}{(n - r + 1)(n - r)!} = n (n - r) { }^{ n } P_{ r - 1 } \] Now, we can express \( { }^{ n } P_{ r - 1 } \): \[ = n (n - r) \cdot \frac{n!}{(n - (r - 1))!} = n (n - r) { }^{ n } P_{ r - 1 } \] Thus, we have shown that: \[ { }^{ n } P_{ r } - { }^{ n } P_{ r - 1 } = (n - r) { }^{ n } P_{ r - 1 } \] This completes the proof.