The number `n` for which number of dissimilar terms in the expansion of `(1+x+x^(2)+x^(3))^(n)` is equal to number of terms in the expansion of `(x+y+z)^(n)` is `(n in N)`________ .

Explanation

To solve the problem, we need to find the natural number \( n \) such that the number of dissimilar terms in the expansion of \( (1 + x + x^2 + x^3)^n \) is equal to the number of terms in the expansion of \( (x + y + z)^n \).

### Step 1: Determine the number of dissimilar terms in \( (1 + x + x^2 + x^3)^n \)

The expression \( 1 + x + x^2 + x^3 \) can be rewritten as a polynomial with terms ranging from \( x^0 \) to \( x^3 \). When raised to the power of \( n \), the highest power of \( x \) in the expansion will be \( 3n \) (since the maximum exponent from \( x^3 \) is multiplied by \( n \)).

The dissimilar terms will be all the powers of \( x \) from \( 0 \) to \( 3n \), which gives us:
\[ \text{Number of dissimilar terms} = 3n + 1 \]

### Step 2: Determine the number of terms in \( (x + y + z)^n \)

Using the multinomial theorem, the number of terms in the expansion of \( (x + y + z)^n \) is given by the formula:
\[ \text{Number of terms} = \binom{n + k - 1}{k - 1} \]
where \( k \) is the number of variables. Here, \( k = 3 \) (for \( x, y, z \)), so we have:
\[ \text{Number of terms} = \binom{n + 3 - 1}{3 - 1} = \binom{n + 2}{2} \]

### Step 3: Set the two expressions equal to each other

We need to set the number of dissimilar terms equal to the number of terms:
\[ 3n + 1 = \binom{n + 2}{2} \]

### Step 4: Expand the binomial coefficient

The binomial coefficient \( \binom{n + 2}{2} \) can be expanded as:
\[ \binom{n + 2}{2} = \frac{(n + 2)(n + 1)}{2} \]

### Step 5: Set up the equation

Now we can rewrite our equation:
\[ 3n + 1 = \frac{(n + 2)(n + 1)}{2} \]

### Step 6: Clear the fraction

Multiply both sides by 2 to eliminate the fraction:
\[ 2(3n + 1) = (n + 2)(n + 1) \]
\[ 6n + 2 = n^2 + 3n + 2 \]

### Step 7: Rearrange the equation

Rearranging gives us:
\[ n^2 + 3n + 2 - 6n - 2 = 0 \]
\[ n^2 - 3n = 0 \]

### Step 8: Factor the equation

Factoring out \( n \):
\[ n(n - 3) = 0 \]

### Step 9: Solve for \( n \)

Setting each factor to zero gives us:
\[ n = 0 \quad \text{or} \quad n - 3 = 0 \Rightarrow n = 3 \]

Since \( n \) must be a natural number, we discard \( n = 0 \) and conclude:
\[ n = 3 \]

### Final Answer
The required value of \( n \) is \( 3 \).

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