Let k and n be positive integers such that 1 ≤ k ≤ n. Verify that the following relations are true:

(i) (n + 1)! = (n + 1)n!;

(ii) (n)1 = n;

(iii) (n)n−1 = n!;

(iv) (n)k = n ⋅ (n − 1)k−1, k ≥ 2;

(v) (n)k = (n − k + 1) ⋅ (n)k−1, 2 ≤ k ≤ n + 1;

(vi) (n)k = (n)r ⋅ (n − r)k−r, 1 ≤ r ≤ k;

(vii) (n)k = n n − k

⋅ (n − 1)k, 1 ≤ k ≤ n − 1.