How many ways are there to place a set of chess pieces on the first row of chessboard. The set consists of a king, a queen, two identical rooks, knights & bishops?

Options

A. A) 8!
B. B) `8^8`
C. C) `5040`
D. D) `4280`

Answer

To solve the problem of arranging the chess pieces on the first row of a chessboard, we will follow these steps:

### Step 1: Identify the pieces
We have the following pieces to arrange:
- 1 King (K)
- 1 Queen (Q)
- 2 identical Rooks (R)
- 2 identical Knights (N)
- 2 identical Bishops (B)

### Step 2: Count the total number of pieces
The total number of pieces is:
- 1 King + 1 Queen + 2 Rooks + 2 Knights + 2 Bishops = 8 pieces.

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of n items where there are groups of identical items is given by:

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

Where:
- \( n \) is the total number of items,
- \( n_1, n_2, \ldots, n_k \) are the counts of each type of identical item.

### Step 4: Apply the formula
In our case:
- Total pieces \( n = 8 \)
- Identical pieces:
- Rooks: 2 (so \( 2! \))
- Knights: 2 (so \( 2! \))
- Bishops: 2 (so \( 2! \))

Thus, the number of arrangements is:

\[ \text{Number of arrangements} = \frac{8!}{2! \times 2! \times 2!} \]

### Step 5: Calculate the factorials
We know:
- \( 8! = 40320 \)
- \( 2! = 2 \)

Now, substituting these values into the formula:

\[ \text{Number of arrangements} = \frac{40320}{2 \times 2 \times 2} = \frac{40320}{8} = 5040 \]

### Conclusion
The total number of ways to arrange the chess pieces on the first row of the chessboard is **5040**.

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