The number of dissimilar terms in the expansion of `(1+x^(4)+2x^(2))^(15)` is

Options

A. 21
B. 31
C. 41
D. 61

Answer

To find the number of dissimilar terms in the expansion of \((1 + x^4 + 2x^2)^{15}\), we can follow these steps:

### Step 1: Identify the terms in the expression
The expression consists of three terms: \(1\), \(x^4\), and \(2x^2\).

### Step 2: Use the multinomial expansion
The expansion of \((a + b + c)^n\) can be expressed using the multinomial theorem. Here, we have:
\[ (1 + x^4 + 2x^2)^{15} \]
We can denote \(a = 1\), \(b = x^4\), and \(c = 2x^2\).

### Step 3: Determine the general term
The general term in the expansion can be expressed as:
\[ \frac{15!}{p!q!r!} (1)^p (x^4)^q (2x^2)^r \]
where \(p + q + r = 15\).

This simplifies to:
\[ \frac{15!}{p!q!r!} \cdot 2^r \cdot x^{4q + 2r} \]

### Step 4: Find the exponent of \(x\)
The exponent of \(x\) in each term is given by:
\[ 4q + 2r \]
We need to find the distinct values of \(4q + 2r\) as \(p\), \(q\), and \(r\) vary under the constraint \(p + q + r = 15\).

### Step 5: Express \(r\) in terms of \(q\)
From \(p + q + r = 15\), we can express \(r\) as:
\[ r = 15 - p - q \]
Substituting \(r\) into the exponent of \(x\):
\[ 4q + 2(15 - p - q) = 4q + 30 - 2p - 2q = 2q + 30 - 2p \]
Thus, the exponent of \(x\) becomes:
\[ 2q - 2p + 30 \]

### Step 6: Determine the range of values
Now we need to find the possible values of \(2q - 2p\). The values of \(p\) and \(q\) can range from \(0\) to \(15\).

- If \(p = 0\), \(q\) can be \(0\) to \(15\), giving \(2q\) values from \(0\) to \(30\).
- If \(p = 15\), \(q\) can be \(0\), giving \(-30\).

Thus, \(2q - 2p\) can take values from \(-30\) to \(30\).

### Step 7: Calculate distinct values
The distinct values of \(2q - 2p\) can be \(0, \pm 2, \pm 4, \ldots, \pm 30\). This gives us \(31\) distinct values (from \(-30\) to \(30\), including \(0\)).

### Conclusion
The number of dissimilar terms in the expansion of \((1 + x^4 + 2x^2)^{15}\) is \(31\).