Question

The total number of permutations of \(n\) different things taken not more than \(r\) at a time, when each thing may be repeated any number of times is

• A. \(\dfrac{n(n^{r+1}-1)}{n-1}\)
• B. \(\dfrac{n^{r+1}-1}{n-1}\)
• C. \(\dfrac{n(n^r-1)}{n-1}\)
• D. \(\dfrac{n^r-1}{n-1}\)

Hint:

When repetition is allowed, the number of arrangements of length \(k\) from \(n\) objects is simply \(n^k\).

Answer 1

The Correct Option is C

Step 1: Understand the meaning of repetition.
Since repetition is allowed, each position in a permutation can be filled in \(n\) ways independently.
Therefore, the number of permutations of length \(k\) is
\[ n^k \]

Step 2: Count permutations for all lengths from \(1\) to \(r\).
The question asks for permutations taken not more than \(r\) at a time.
Hence, total number of permutations is
\[ n^1+n^2+n^3+\cdots+n^r \]

Step 3: Use the geometric progression formula.
The above series is a geometric progression with:
\[ a=n,\qquad \text{common ratio}=n \]
Sum of GP with \(r\) terms is
\[ S=\frac{a(r^n-1)}{r-1} \] Using the correct GP formula here,
\[ S=\frac{n(n^r-1)}{n-1} \]
Thus, the required number of permutations is
\[ \frac{n(n^r-1)}{n-1} \]

Step 4: Final conclusion.
Hence,
\[ \boxed{\frac{n(n^r-1)}{n-1}} \]