Answer to Question #204897 in Discrete Mathematics for Raj

Question

How many distinguishable permutations with repeated elements are there for the number 20366650?

Answer 1

The number of permutations of $n$ elements taken $n$ at a time, with $r_1$ elements of one kind, $r_2$ elements of another kind, and so on, is

$$\dfrac{n!}{r_1!r_2!...r_k!}$$

20366650

There are $n=8$ digits. The digit repeats $r_1=2$ times, the digit $6$ repeats $r_2=3$ times.

Then

$n=8, r_1=2, r_2=3$

$$\dfrac{8!}{2!3!}=\dfrac{8(7)(6)(5)(4)}{1(2)(3)}=1120$$