If the number of terms in the expansion of `(x+y+z)^n` are 36, then find the value of `ndot`

Explanation

To find the value of \( n \) given that the number of terms in the expansion of \( (x+y+z)^n \) is 36, we can follow these steps:

### Step 1: Understand the formula for the number of terms
The number of terms in the expansion of \( (x_1 + x_2 + x_3 + \ldots + x_r)^n \) is given by the formula:

\[ \text{Number of terms} = \binom{n + r - 1}{r - 1} \]

where \( r \) is the number of different variables (in our case, \( x, y, z \)), so \( r = 3 \).

### Step 2: Substitute the values into the formula
For our problem, we have:

\[ \text{Number of terms} = \binom{n + 3 - 1}{3 - 1} = \binom{n + 2}{2} \]

We know from the problem statement that this equals 36:

\[ \binom{n + 2}{2} = 36 \]

### Step 3: Expand the binomial coefficient
The binomial coefficient can be expanded as follows:

\[ \binom{n + 2}{2} = \frac{(n + 2)(n + 1)}{2} \]

### Step 4: Set up the equation
Setting the expanded binomial coefficient equal to 36 gives us:

\[ \frac{(n + 2)(n + 1)}{2} = 36 \]

### Step 5: Multiply both sides by 2
To eliminate the fraction, multiply both sides by 2:

\[ (n + 2)(n + 1) = 72 \]

### Step 6: Expand the left side
Expanding the left side yields:

\[ n^2 + 3n + 2 = 72 \]

### Step 7: Rearrange the equation
Rearranging the equation gives:

\[ n^2 + 3n + 2 - 72 = 0 \]

which simplifies to:

\[ n^2 + 3n - 70 = 0 \]

### Step 8: Factor the quadratic equation
Now we need to factor the quadratic equation:

\[ (n + 10)(n - 7) = 0 \]

### Step 9: Solve for \( n \)
Setting each factor to zero gives us:

\[ n + 10 = 0 \quad \Rightarrow \quad n = -10 \quad (\text{not valid since } n \text{ must be non-negative}) \]
\[ n - 7 = 0 \quad \Rightarrow \quad n = 7 \]

### Conclusion
The value of \( n \) is:

\[ \boxed{7} \]