How many permutations can be there with `x^(3).y^(2) z^(4)`? How many of these arrangements, all z are not together?

Answer

To solve the problem of finding the number of permutations of the expression \(x^3 y^2 z^4\) and the arrangements where all \(z\)s are not together, we can follow these steps:

### Step 1: Calculate the total number of permutations

The total number of permutations of the letters in the expression \(x^3 y^2 z^4\) can be calculated using the formula for permutations of multiset:

\[ \text{Total permutations} = \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \]

Where:
- \(n\) is the total number of items,
- \(n_1, n_2, n_3\) are the counts of each distinct item.

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

Thus, we have:

\[ \text{Total permutations} = \frac{9!}{3! \cdot 2! \cdot 4!} \]

### Step 2: Calculate the factorials

Now we compute the factorials:

- \(9! = 362880\)
- \(3! = 6\)
- \(2! = 2\)
- \(4! = 24\)

### Step 3: Substitute the values into the formula

Now substituting these values into the formula:

\[ \text{Total permutations} = \frac{362880}{6 \cdot 2 \cdot 24} \]

Calculating the denominator:

\[ 6 \cdot 2 = 12 \]
\[ 12 \cdot 24 = 288 \]

Now we can compute:

\[ \text{Total permutations} = \frac{362880}{288} = 1260 \]

### Step 4: Calculate the arrangements where all \(z\)s are together

To find the arrangements where all \(z\)s are together, we can treat all \(z\)s as a single unit or "block". Thus, we have:

- One block of \(z\)s,
- Three \(x\)s,
- Two \(y\)s.

This gives us a total of \(3 + 2 + 1 = 6\) units to arrange.

The number of arrangements of these units is given by:

\[ \text{Arrangements with all } z \text{ together} = \frac{6!}{3! \cdot 2!} \]

### Step 5: Calculate the factorials for this arrangement

Calculating \(6!\):

- \(6! = 720\)

Now substituting into the formula:

\[ \text{Arrangements with all } z \text{ together} = \frac{720}{6 \cdot 2} = \frac{720}{12} = 60 \]

### Step 6: Calculate the arrangements where no \(z\)s are together

To find the arrangements where no \(z\)s are together, we subtract the arrangements where all \(z\)s are together from the total arrangements:

\[ \text{Arrangements where no } z \text{ is together} = \text{Total permutations} - \text{Arrangements with all } z \text{ together} \]

Substituting the values:

\[ \text{Arrangements where no } z \text{ is together} = 1260 - 60 = 1200 \]

### Final Answers

1. The total number of permutations is **1260**.
2. The number of arrangements where no \(z\)s are together is **1200**.