The expansion of `(x + y + z)^(n)` is given by

Options

A. `sum (n!)/(p!q!r!)x^(p)q^(q)z^(r)` where p,q,r`ge0,p+q+r=n`
B. `sum (n!)/(p!q!r!)x^(p)q^(q)z^(r)` where p+q+r=n
C. `sum ((n+1)!)/(p!q!r!)x^(p)q^(q)z^(r)` where p+q+r=n
D. none of these

Answer

To find the expansion of \((x + y + z)^{n}\) using the Multinomial Theorem, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Multinomial Theorem**:
The Multinomial Theorem states that the expansion of \((x_1 + x_2 + ... + x_k)^n\) can be expressed in terms of the coefficients and powers of each term. For three variables \(x\), \(y\), and \(z\), the expansion is given by:
\[ (x + y + z)^n = \sum_{p + q + r = n} \frac{n!}{p! q! r!} x^p y^q z^r \]
where \(p\), \(q\), and \(r\) are non-negative integers.

2. **Identifying the General Term**:
In the expansion, each term corresponds to a specific combination of powers of \(x\), \(y\), and \(z\). The general term can be written as:
\[ T_{p,q,r} = \frac{n!}{p! q! r!} x^p y^q z^r \]
Here, \(p\), \(q\), and \(r\) represent the powers of \(x\), \(y\), and \(z\) respectively.

3. **Condition for the Powers**:
The powers \(p\), \(q\), and \(r\) must satisfy the condition:
\[ p + q + r = n \]
Additionally, \(p\), \(q\), and \(r\) must be non-negative integers, which means:
\[ p, q, r \geq 0 \]

4. **Counting the Number of Terms**:
The total number of distinct terms in the expansion can be found using the combinatorial formula for distributing \(n\) identical items (the total degree) into \(k\) distinct groups (the variables). This is given by:
\[ \text{Number of terms} = \binom{n + k - 1}{k - 1} \]
For our case with \(k = 3\) (for \(x\), \(y\), and \(z\)):
\[ \text{Number of terms} = \binom{n + 3 - 1}{3 - 1} = \binom{n + 2}{2} \]

5. **Conclusion**:
The expansion of \((x + y + z)^n\) is thus given by the sum of all terms \(T_{p,q,r}\) where \(p + q + r = n\) and \(p, q, r \geq 0\).

### Final Expression:
The final expression for the expansion is:
\[ (x + y + z)^n = \sum_{p + q + r = n} \frac{n!}{p! q! r!} x^p y^q z^r \]