The number of six digit numbers that can be formed by using the digits 1,2,1,2,0,2 is

Options

A. 50
B. 60
C. 110
D. 10

Answer

To find the number of six-digit numbers that can be formed using the digits 1, 2, 1, 2, 0, and 2, we can follow these steps:

### Step 1: Identify the digits and their occurrences
The digits we have are: 1, 1, 2, 2, 2, 0.
- The digit 1 appears 2 times.
- The digit 2 appears 3 times.
- The digit 0 appears 1 time.

### Step 2: Determine the restrictions for six-digit numbers
A six-digit number cannot start with the digit 0. Therefore, we can only use the digits 1 or 2 in the first position.

### Step 3: Case analysis based on the first digit
We will analyze two cases based on the first digit.

#### Case 1: First digit is 1
If the first digit is 1, the remaining digits are 1, 2, 2, 2, and 0.
- The total digits left: 1, 2, 2, 2, 0 (5 digits)
- The number of arrangements of these digits can be calculated using the formula for permutations of multiset:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \cdot p_2! \cdot \ldots} \]

Where \( n \) is the total number of items, and \( p_1, p_2, \ldots \) are the counts of each distinct item.

Here, we have:
- Total digits = 5 (1, 2, 2, 2, 0)
- The digit 2 appears 3 times.

Thus, the number of arrangements is:

\[ \text{Arrangements} = \frac{5!}{3! \cdot 1!} = \frac{120}{6} = 20 \]

#### Case 2: First digit is 2
If the first digit is 2, the remaining digits are 1, 1, 2, 2, and 0.
- The total digits left: 1, 1, 2, 2, 0 (5 digits)
- The number of arrangements of these digits is calculated similarly:

Here, we have:
- Total digits = 5 (1, 1, 2, 2, 0)
- The digit 1 appears 2 times and the digit 2 appears 2 times.

Thus, the number of arrangements is:

\[ \text{Arrangements} = \frac{5!}{2! \cdot 2!} = \frac{120}{4} = 30 \]

### Step 4: Combine the results from both cases
Now, we add the number of arrangements from both cases to find the total number of six-digit numbers:

\[ \text{Total} = \text{Case 1} + \text{Case 2} = 20 + 30 = 50 \]

### Final Answer
The total number of six-digit numbers that can be formed is **50**.

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