The symbols ` + , + , xx , xx , *** ,cdot` are placed in the squares of the adjoining figure. The number of ways of placing symbols so that no row remains empty is

Options

A. 16580
B. 14580
C. 5400
D. 4860

Answer

To solve the problem of placing the symbols ` + , + , xx , xx , *** , cdot` in such a way that no row remains empty, we can follow these steps:

### Step 1: Identify the Symbols and Rows
We have the following symbols:
- 2 `+` symbols
- 2 `xx` symbols
- 3 `*` symbols
- 1 `cdot` symbol

We need to place these symbols in 5 rows, ensuring that no row is empty.

### Step 2: Choose Symbols for Each Row
Since we have 5 rows and we need to ensure that each row has at least one symbol, we can start by placing one symbol in each of the 5 rows.

### Step 3: Assign Symbols to Rows
We can choose 5 symbols from the available symbols to fill the rows. The choices can be made as follows:
- Choose 1 symbol from the `+` symbols (2 choices)
- Choose 1 symbol from the `xx` symbols (2 choices)
- Choose 1 symbol from the `*` symbols (3 choices)
- Choose 1 symbol from the `cdot` symbol (1 choice)

This gives us:
- 2 choices for `+`
- 2 choices for `xx`
- 3 choices for `*`
- 1 choice for `cdot`

Thus, the total number of ways to choose one symbol for each row is:
\[ 2 \times 2 \times 3 \times 1 = 12 \]

### Step 4: Choose the Remaining Symbols
After placing one symbol in each of the 5 rows, we have 6 symbols left (3 `*` and 2 `xx` and 1 `cdot`). We need to choose one of these symbols to fill one more row.

The number of ways to choose one symbol from the remaining 6 symbols is:
\[ \binom{6}{1} = 6 \]

### Step 5: Calculate Total Arrangements
Now we need to calculate the total arrangements of the symbols. The total number of arrangements of the symbols is given by:
\[ \frac{6!}{2! \times 2! \times 3!} \]

Where:
- \(6!\) is the factorial of the total number of symbols.
- \(2!\) accounts for the identical `+` symbols.
- \(2!\) accounts for the identical `xx` symbols.
- \(3!\) accounts for the identical `*` symbols.

Calculating this gives:
\[ \frac{720}{2 \times 2 \times 6} = \frac{720}{24} = 30 \]

### Step 6: Combine Choices and Arrangements
Finally, we multiply the number of ways to choose the symbols by the number of arrangements:
\[ 12 \times 30 = 360 \]

### Final Answer
The total number of ways to place the symbols in the squares so that no row remains empty is **360**.

---