Question

The number of permutations of 20 dissimilar things taken not more than 10 at a time when repetitions are allowed is

• A. $\frac{20}{19}({20}^{20}-1)$
• B. $\frac{20}{19}({20}^{10}+1)$
• C. $\frac{20}{19}({20}^{10}-1)$
• D. $\frac{20}{19}(2^{20}+1)$

Answer 1

The correct option is C $\frac{20}{19}({20}^{10}-1)$

Let us find the number of permutations for selecting $r$ objects $(r\le 10)$ from 20 dissimilar objects.
Since, repetitions are allowed, each object can be selected in 20 ways.
Hence, total number of permutations $={20}^r$.
We can vary $r$ from 1 to 10.
Hence, total number of permutations $=20+{20}^2+...+{20}^{10}=\frac{20}{19}({20}^{10}-1)$
This is a geometric series, with $a=20$, $r=20$, $n=10$
Hence,
Number of Permutations $=a\frac{r^n-1}{r-1}$
Number of Permutations $=20\left(\frac{{20}^{10}-1}{19}\right)$
Hence, option C is correct.