Animated Solution for Mathematics - Probability: The probability that a randomly chosen 5-digit number is made from exactly two digits is :

To solve this problem, we first define our sample space, $n(S)$. A 5-digit number is represented by the slots $d_1{d}_2{d}_3{d}_4{d}_5$.

The first digit $d_1$ cannot be $0$, leaving $9$ choices ($1$ through $9$). The remaining four slots can each be any of the $10$ digits ($0$ through $9$).

This value represents the total number of possible 5-digit numbers.

We must form numbers using exactly two distinct digits. Because $0$ cannot occupy the first position, we must split our analysis into two mutually exclusive cases to avoid counting errors.

The Zero-Included Scenario

Suppose our two digits are $0$ and a non-zero digit $x$. There are $9$ possible choices for $x$ ($\{1,2,\dots ,9\}$).

For the arrangement, the first slot $d_1$ must be $x$ (1 choice). The remaining four slots can be either $0$ or $x$, yielding $2^4=16$ raw combinations.

We must exclude the case where all four remaining slots are $x$, as that would result in a 1-digit number ($xxxxx$). Thus, we have $16-1=15$ valid arrangements per pair.

The Non-Zero Scenario

Now, consider the case where both chosen digits $x$ and $y$ are non-zero. We choose $2$ digits from $9$, which is given by:

Since neither digit is zero, there are no restrictions on $d_1$. Each of the $5$ slots can be either $x$ or $y$, giving $2^5=32$ raw arrangements.

We must exclude the two cases where the number consists of only one digit (all $x$'s or all $y$'s). This leaves $32-2=30$ valid arrangements per pair.

The total number of favorable outcomes $n(E)$ is the sum of our two cases:

The probability $P(E)$ is the ratio of favorable outcomes to the total sample space:

Dividing both the numerator and denominator by $135$, we arrive at the final simplified probability: