‘n’ digit positive integers formed such that each digit is 1, 2, or 3. How many of these contain all three of the digits 1, 2, and 3 at-least once?

Options

A. `3(n-1)`
B. `3^(n)-2*2^(n)+3`
C. `3^(n)-3*2^(n)-3`
D. `3^(n)-3*2^(n)+3`

Answer

To solve the problem of finding how many n-digit positive integers can be formed using the digits 1, 2, and 3 such that each digit appears at least once, we can follow these steps:

### Step 1: Calculate the Total Number of n-Digit Integers
The total number of n-digit integers that can be formed using the digits 1, 2, and 3 is given by:
\[ \text{Total integers} = 3^n \]
This is because for each of the n positions in the integer, we have 3 choices (1, 2, or 3).

**Hint:** Remember that for each digit in the number, you can choose any of the three digits independently.

### Step 2: Calculate the Number of Integers Missing at Least One Digit
To find the number of integers that contain at least one of the digits 1, 2, or 3, we can use the principle of complementary counting. We will first calculate the number of integers that are missing at least one digit.

#### Step 2.1: Count Integers with Only Two Digits
The number of integers that can be formed using only two of the three digits can be calculated as follows:
- Choose 2 digits from 3: \( \binom{3}{2} = 3 \)
- For each choice of 2 digits, the number of n-digit integers that can be formed is \( 2^n \).

Thus, the total number of integers that use only two digits is:
\[ \text{Integers with 2 digits} = 3 \times 2^n \]

**Hint:** Use combinations to determine how many ways you can choose the digits.

#### Step 2.2: Count Integers with Only One Digit
The number of integers that can be formed using only one digit is simply:
\[ \text{Integers with 1 digit} = 3 \quad (\text{1, 2, or 3}) \]

**Hint:** There are only three choices when you restrict yourself to one digit.

### Step 3: Apply the Inclusion-Exclusion Principle
Now, we can calculate the total number of integers that are missing at least one digit:
\[ \text{Integers missing at least one digit} = \text{Integers with 2 digits} + \text{Integers with 1 digit} \]
\[ = 3 \times 2^n + 3 \]

### Step 4: Calculate the Number of Integers Containing All Three Digits
Finally, we can find the number of n-digit integers that contain all three digits by subtracting the number of integers missing at least one digit from the total number of integers:
\[ \text{Integers with all three digits} = 3^n - (3 \times 2^n + 3) \]
\[ = 3^n - 3 \times 2^n - 3 \]

### Final Answer
Thus, the number of n-digit positive integers formed such that each digit is 1, 2, or 3 and contains all three digits at least once is:
\[ \boxed{3^n - 3 \times 2^n - 3} \]