How many different six-digit numbers can be formed using all of the following digits : 3,3,4,4,4,5.

Options

A. 10
B. 20
C. 30
D. 60

Answer

To find out how many different six-digit numbers can be formed using the digits 3, 3, 4, 4, 4, and 5, we can use the formula for permutations of multiset.

### Step-by-Step Solution:

1. **Identify the Total Digits**:
We have the digits: 3, 3, 4, 4, 4, 5.
This gives us a total of 6 digits.

2. **Count the Frequency of Each Digit**:
- The digit 3 appears 2 times.
- The digit 4 appears 3 times.
- The digit 5 appears 1 time.

3. **Use the Permutation Formula for Multisets**:
The formula for the number of arrangements of n objects where there are n1 identical objects of one type, n2 identical objects of another type, and so on, is given by:
\[ \text{Number of arrangements} = \frac{n!}{n1! \times n2! \times ...} \]
In our case:
- \( n = 6 \) (total digits)
- \( n1 = 2 \) (for the two 3's)
- \( n2 = 3 \) (for the three 4's)
- \( n3 = 1 \) (for the one 5)

Thus, we can write:
\[ \text{Number of arrangements} = \frac{6!}{2! \times 3! \times 1!} \]

4. **Calculate Factorials**:
- \( 6! = 720 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 1! = 1 \)

5. **Substitute the Values into the Formula**:
\[ \text{Number of arrangements} = \frac{720}{2 \times 6 \times 1} = \frac{720}{12} \]

6. **Perform the Division**:
\[ \frac{720}{12} = 60 \]

7. **Conclusion**:
Therefore, the total number of different six-digit numbers that can be formed using the digits 3, 3, 4, 4, 4, and 5 is **60**.