How many three-digit number can be generated from 1, 2, 3, 4, 5, 6, 7, 8, 9, such that the digits are in ascending order?

Options

A. 80
B. 81
C. 83
D. 84

Answer

To solve the problem of how many three-digit numbers can be generated from the digits 1 to 9 such that the digits are in ascending order, we can follow these steps:

### Step 1: Understand the Problem
We need to form three-digit numbers using the digits from 1 to 9. The key condition is that the digits must be in ascending order. This means that if we choose any three digits, they will automatically form a valid number when arranged in ascending order.

### Step 2: Choose the Digits
Since the digits must be in ascending order, we can simply choose any three digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. The order of selection does not matter because they will be arranged in ascending order.

### Step 3: Calculate the Combinations
To find the number of ways to choose 3 digits from 9, we use the combination formula:

\[ \text{Number of combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In our case, \(n = 9\) (the total digits) and \(r = 3\) (the digits we want to choose):

\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3! \cdot 6!} \]

### Step 4: Simplify the Calculation
We can simplify the calculation:

\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \]

Calculating the numerator:

\[ 9 \times 8 = 72 \]
\[ 72 \times 7 = 504 \]

Now, calculating the denominator:

\[ 3 \times 2 \times 1 = 6 \]

Now, divide the numerator by the denominator:

\[ \frac{504}{6} = 84 \]

### Step 5: Conclusion
Thus, the total number of three-digit numbers that can be formed from the digits 1 to 9 in ascending order is **84**.