The number of 3-digit numbers which consists of the digits in A.P., strictly in increasing order using the non-zero digits of the decimal system is :

Options

A. 14
B. 16
C. 15
D. none of these

Answer

To find the number of 3-digit numbers consisting of digits in an arithmetic progression (A.P.) that are strictly in increasing order using the non-zero digits of the decimal system, we can follow these steps:

### Step 1: Understand the Conditions
We need to form a 3-digit number represented as \( xyz \), where:
- \( x, y, z \) are the digits.
- The digits must be in strictly increasing order: \( x < y < z \).
- The digits must be in A.P., which means \( 2y = x + z \).

### Step 2: Identify Possible Digits
The non-zero digits of the decimal system are \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \).

### Step 3: Set Up the A.P. Condition
From the A.P. condition \( 2y = x + z \), we can rearrange it to find \( y \):
\[ y = \frac{x + z}{2} \]
This means \( x + z \) must be even for \( y \) to be an integer.

### Step 4: List Possible Combinations
We will consider pairs \( (x, z) \) such that \( x < z \) and \( x + z \) is even. The possible pairs of \( (x, z) \) are:
- \( (1, 3) \) → \( y = 2 \)
- \( (1, 5) \) → \( y = 3 \)
- \( (1, 7) \) → \( y = 4 \)
- \( (1, 9) \) → \( y = 5 \)
- \( (2, 4) \) → \( y = 3 \)
- \( (2, 6) \) → \( y = 4 \)
- \( (2, 8) \) → \( y = 5 \)
- \( (3, 5) \) → \( y = 4 \)
- \( (3, 7) \) → \( y = 5 \)
- \( (4, 6) \) → \( y = 5 \)

### Step 5: Verify Each Combination
Now we will check which combinations yield valid 3-digit numbers:
1. \( (1, 3) \) → \( 123 \)
2. \( (1, 5) \) → \( 135 \)
3. \( (1, 7) \) → \( 147 \)
4. \( (1, 9) \) → \( 159 \)
5. \( (2, 4) \) → \( 246 \)
6. \( (2, 6) \) → \( 258 \)
7. \( (2, 8) \) → \( 268 \)
8. \( (3, 5) \) → \( 357 \)
9. \( (3, 7) \) → \( 369 \)
10. \( (4, 6) \) → \( 468 \)

### Step 6: Count the Valid Combinations
From the valid combinations, we have the following 3-digit numbers:
- 123
- 135
- 147
- 159
- 246
- 258
- 268
- 357
- 369
- 468

Thus, we have a total of **10 valid 3-digit numbers**.

### Conclusion
The number of 3-digit numbers which consist of the digits in A.P., strictly in increasing order using the non-zero digits of the decimal system is **10**.

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