What is the number of different messages that can be represented by three 0's and two 1's ?

Options

A. 10
B. 9
C. 8
D. 7

Answer

To find the number of different messages that can be represented by three 0's and two 1's, we can use the concept of permutations of multiset. Here are the steps to solve the problem:

### Step 1: Identify the total number of symbols
We have a total of 5 symbols: three 0's and two 1's.

### Step 2: Use the formula for permutations of multiset
The formula to find the number of distinct arrangements of n items where there are groups of identical items is given by:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2!} \]

Where:
- \( n \) is the total number of items,
- \( n_1 \) is the number of identical items of one type (0's),
- \( n_2 \) is the number of identical items of another type (1's).

### Step 3: Substitute the values into the formula
In our case:
- \( n = 5 \) (total symbols),
- \( n_1 = 3 \) (for the three 0's),
- \( n_2 = 2 \) (for the two 1's).

So, we can substitute these values into the formula:

\[ \text{Number of arrangements} = \frac{5!}{3! \times 2!} \]

### Step 4: Calculate the factorials
Now, we calculate the factorials:
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 5: Substitute the factorial values back into the equation
Now we substitute these values back into the equation:

\[ \text{Number of arrangements} = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \]

### Step 6: Conclusion
Thus, the number of different messages that can be represented by three 0's and two 1's is **10**.