How many different numbers can be formed out of all the digits 111223?

The exact answer to the question "How many different numbers can be formed out of all the digits 111223?" is 60.

Understanding Permutations with Repetition

When forming numbers from a set of digits where some digits are repeated, the number of unique arrangements (permutations) is calculated using a specific formula. This is different from permutations where all items are distinct. The question implies forming six-digit numbers by using all the provided digits.

The formula for permutations with repetition is:

$$ \frac{n!}{n_1! n_2! \dots n_k!} $$

Where:

• $n$ is the total number of items (digits) available.
• $n_1, n_2, \dots, n_k$ are the counts of each distinct item that is repeated.

Applying the Formula to the Digits 111223

Let's break down the given digits:

• Total number of digits ($n$) = 6
• Digit '1' appears ($n_1$) = 3 times
• Digit '2' appears ($n_2$) = 2 times
• Digit '3' appears ($n_3$) = 1 time

Now, we can apply the formula:

1. Calculate the factorial of the total number of digits:
   $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

2. Calculate the factorial for each repeated digit's count:

   • For '1' (3 times): $3! = 3 \times 2 \times 1 = 6$
   • For '2' (2 times): $2! = 2 \times 1 = 2$
   • For '3' (1 time): $1! = 1$ (This is included for completeness but doesn't change the product).

3. Divide the total factorial by the product of the factorials of the repeated counts:
   Number of unique numbers = $ \frac{6!}{3! \times 2! \times 1!} = \frac{720}{6 \times 2 \times 1} = \frac{720}{12} = 60 $

Therefore, there are 60 unique six-digit integers that can be formed from the digits 111223.

Practical Insights

This method is crucial for problems involving arrangements of items with identical elements. For instance, it's used in:

• Calculating the number of unique words that can be formed from a set of letters (e.g., how many unique words can be formed from the letters of "MISSISSIPPI").
• Determining unique sequences in genetic codes or data strings where certain characters repeat.

Summary of Digits and Counts:

The calculation confirms that there are 60 different numbers that can be formed using all the digits 111223.