How many unequal 5 -digit numbers can be formed using each digit of the number 11235 only once?

Options

A. `5!`
B. `._(5)P_(3)`
C. `(._(5)C_5)/(2!)`
D. `(._(5)P_(5))/(2! . 3!)`

Answer

To find the number of unequal 5-digit numbers that can be formed using each digit of the number 11235 only once, we can follow these steps:

### Step 1: Identify the digits and their frequencies
The digits in the number 11235 are:
- 1 (occurs 2 times)
- 2 (occurs 1 time)
- 3 (occurs 1 time)
- 5 (occurs 1 time)

### Step 2: Use the formula for permutations of multiset
When we have a set of items where some items are identical, the number of distinct arrangements can be calculated using the formula:
\[ \frac{n!}{n_1! \times n_2! \times \ldots} \]
where:
- \( n \) is the total number of items,
- \( n_1, n_2, \ldots \) are the frequencies of the identical items.

In our case:
- Total digits \( n = 5 \) (1, 1, 2, 3, 5)
- The digit '1' appears 2 times, and the other digits appear 1 time each.

### Step 3: Substitute into the formula
Using the formula, we have:
\[ \text{Number of distinct arrangements} = \frac{5!}{2!} \]

### Step 4: Calculate the factorials
Now, we calculate the factorials:
- \( 5! = 120 \)
- \( 2! = 2 \)

### Step 5: Perform the division
Now, we can perform the division:
\[ \frac{120}{2} = 60 \]

### Conclusion
Thus, the total number of unequal 5-digit numbers that can be formed using each digit of the number 11235 only once is **60**.

---