If the number of terms in the expansion `(1+2x-3y+4z)^(n)` is `286`, then find the coefficient of term containing `xyz`.

Explanation

To solve the problem step by step, we need to find the coefficient of the term containing \(xyz\) in the expansion of \((1 + 2x - 3y + 4z)^n\) given that the number of terms in the expansion is 286.

### Step 1: Determine the number of terms in the expansion
The number of terms in the expansion of \((a_1 + a_2 + a_3 + a_4)^n\) is given by the formula:
\[ \text{Number of terms} = \binom{n + k - 1}{k - 1} \]
where \(k\) is the number of different terms. Here, we have \(k = 4\) (the terms are \(1\), \(2x\), \(-3y\), and \(4z\)). Thus, the number of terms is:
\[ \binom{n + 4 - 1}{4 - 1} = \binom{n + 3}{3} \]
According to the problem, this is equal to 286:
\[ \binom{n + 3}{3} = 286 \]

### Step 2: Solve for \(n\)
Using the formula for combinations:
\[ \binom{n + 3}{3} = \frac{(n + 3)(n + 2)(n + 1)}{3!} = \frac{(n + 3)(n + 2)(n + 1)}{6} \]
Setting this equal to 286, we have:
\[ \frac{(n + 3)(n + 2)(n + 1)}{6} = 286 \]
Multiplying both sides by 6 gives:
\[ (n + 3)(n + 2)(n + 1) = 1716 \]

### Step 3: Factor and solve the equation
Now we need to find \(n\) such that:
\[ (n + 3)(n + 2)(n + 1) = 1716 \]
We can try different integer values for \(n\). Testing \(n = 10\):
\[ (10 + 3)(10 + 2)(10 + 1) = 13 \cdot 12 \cdot 11 = 1716 \]
Thus, \(n = 10\).

### Step 4: Find the coefficient of the term containing \(xyz\)
The term containing \(xyz\) in the expansion can be found using the multinomial expansion. The general term in the expansion can be expressed as:
\[ \frac{n!}{k_1! k_2! k_3! k_4!} (1)^{k_1} (2x)^{k_2} (-3y)^{k_3} (4z)^{k_4} \]
where \(k_1 + k_2 + k_3 + k_4 = n\).

For the term containing \(xyz\), we set \(k_2 = 1\), \(k_3 = 1\), \(k_4 = 1\) and \(k_1 = n - 3\):
\[ k_1 = 10 - 3 = 7 \]
Thus, the coefficient of the term containing \(xyz\) is:
\[ \frac{10!}{7! \cdot 1! \cdot 1! \cdot 1!} (1)^7 (2x)^1 (-3y)^1 (4z)^1 \]
Calculating this gives:
\[ \frac{10!}{7! \cdot 1 \cdot 1 \cdot 1} = \frac{10 \cdot 9 \cdot 8}{1 \cdot 1 \cdot 1} = 720 \]
Now, substituting the values:
\[ 720 \cdot 2 \cdot (-3) \cdot 4 = 720 \cdot 2 \cdot -12 = 720 \cdot -24 = -17280 \]

### Final Answer
The coefficient of the term containing \(xyz\) is \(-17280\).