Distinct Digits Function

Algebra Level 2

The function $\text{dd}(n)$ takes in a positive integer $n$ , and gives the number of distinct digits in $n$ , when written in base 10.

How many elements are there in the range of $\text{dd}(n)$ ?

As an explicit example, $\text{dd}(1123) = 3$ , since there are three distinct digits in 1123, namely 1, 2, and 3.

Infinitely many 11 10 8 9

1 solution

Hung Woei Neoh
May 16, 2016

$\text{dd}(n) =$ number of distinct digits in $n$ at base $10$

At base $10$ , we have $10$ different types of digits: $0,1,2,3,4,5,6,7,8,9$

Each number must have at least $1$ distinct digit, and it will have at most $10$ distinct digits. A number can have $2$ to $9$ distinct digits too.

Therefore, the range of $\text{dd}(n), R_{\text{dd}} = \{1,2,3,4,5,6,7,8,9,10\}$

The number of elements in the range, $n(R_{\text{dd}}) = \boxed{10}$