The total number of arrangements of the letters in the expression `x^3 z^2 y^4` when written at full lengths is

Options

A. 2520
B. 1260
C. 610
D. none of these

Answer

To find the total number of arrangements of the letters in the expression \( x^3 z^2 y^4 \) when written at full lengths, we can follow these steps:

### Step 1: Write the expression in full length
The expression \( x^3 z^2 y^4 \) means we have:
- 3 letters 'x'
- 2 letters 'z'
- 4 letters 'y'

When written out fully, it looks like this:
\[ xxxzzyyyy \]

### Step 2: Count the total number of letters
Now, we count the total number of letters:
- Total letters = 3 (from x) + 2 (from z) + 4 (from y) = 9 letters.

### Step 3: Use the formula for arrangements of letters
The formula to find the number of arrangements of letters when some letters are repeated is given by:
\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3!} \]
where:
- \( n \) is the total number of letters,
- \( n_1, n_2, n_3 \) are the frequencies of the repeated letters.

In our case:
- \( n = 9 \) (total letters),
- \( n_1 = 3 \) (for x),
- \( n_2 = 2 \) (for z),
- \( n_3 = 4 \) (for y).

### Step 4: Substitute values into the formula
Substituting the values into the formula, we get:
\[ \text{Total arrangements} = \frac{9!}{3! \times 2! \times 4!} \]

### Step 5: Calculate the factorials
Now, we calculate the factorials:
- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 2! = 2 \times 1 = 2 \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

### Step 6: Substitute the factorials back into the equation
Now substituting these values back:
\[ \text{Total arrangements} = \frac{362880}{6 \times 2 \times 24} \]

### Step 7: Calculate the denominator
Calculating the denominator:
\[ 6 \times 2 = 12 \]
\[ 12 \times 24 = 288 \]

### Step 8: Final calculation
Now we can calculate the total arrangements:
\[ \text{Total arrangements} = \frac{362880}{288} = 1260 \]

Thus, the total number of arrangements of the letters in the expression \( x^3 z^2 y^4 \) when written at full lengths is **1260**.