A three digit number is chosen, what is the probability of having digits in increasing order from left to right .

Answer

To find the probability of selecting a three-digit number with its digits in increasing order from left to right, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Number of Three-Digit Numbers:**
The range of three-digit numbers is from 100 to 999.
- Total three-digit numbers = 999 - 100 + 1 = 900.

2. **Identify Conditions for Increasing Order:**
A three-digit number has digits in increasing order if each digit is less than the next. For example, in the number 123, 1 < 2 < 3.

3. **Choose Digits for the Increasing Order:**
The digits of a three-digit number can only be chosen from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} because:
- The first digit cannot be 0 (as it would not be a three-digit number).
- The digits must be distinct since they are in increasing order.

4. **Calculate the Number of Ways to Choose 3 Digits:**
We need to select 3 different digits from the 9 available digits (1 to 9). The number of ways to choose 3 digits from 9 is given by the combination formula:
\[ \text{Number of ways} = \binom{9}{3} \]
This can be calculated as:
\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84. \]

5. **Calculate the Probability:**
The probability of selecting a three-digit number with digits in increasing order is the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Number of increasing order numbers}}{\text{Total three-digit numbers}} = \frac{84}{900}. \]
Simplifying this fraction:
\[ \frac{84}{900} = \frac{7}{75}. \]

### Final Answer:
The probability of having the digits of a three-digit number in increasing order from left to right is \( \frac{7}{75} \).