Discussion
If n and r are non-negative integers and \(n \ge r\), then \(p(n + 1, r)\) equals to
Answer: A
The number of permutations is p(n, r) = n! / (n - r)!. For n + 1 objects it is p(n + 1, r) = (n + 1)! / (n + 1 - r)!. Writing (n + 1)! = (n + 1) n! and (n + 1 - r)! = (n + 1 - r)(n - r)! gives the result.
\[ p(n+1, r) = \frac{(n+1)\, n!}{(n+1-r)(n-r)!} \]\[ = \frac{p(n, r)(n+1)}{n+1-r} \]So the first option is right. The other options change the sign of 1 or r in the denominator or the factor n + 1 in the numerator, which does not agree with the derivation.
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