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

Options

A. 21
B. 31
C. 42
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: Rewrite the expression
We start with the expression:
\[ (1 + x^4 - 2x^2)^{15} \]
This can be rewritten as:
\[ (1 + x^4 + (-2x^2))^{15} \]

### Step 2: Identify the terms
Using the binomial theorem, the expansion of \( (a + b + c)^n \) can be expressed as:
\[ \sum \frac{n!}{p!q!r!} a^p b^q c^r \]
where \( p + q + r = n \). In our case, \( a = 1 \), \( b = x^4 \), and \( c = -2x^2 \).

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

### Step 4: Find the exponent of \( x \)
The exponent of \( x \) in each term is given by:
\[ 4q + 2r \]
To find the number of dissimilar terms, we need to find the possible values of \( 4q + 2r \).

### Step 5: Set up the constraints
From \( p + q + r = 15 \), we can express \( p \) in terms of \( q \) and \( r \):
\[ p = 15 - q - r \]
Now we need to find the range of \( 4q + 2r \).

### Step 6: Analyze the values of \( q \) and \( r \)
1. The maximum value of \( q \) is 15 (when \( r = 0 \)), giving \( 4q = 60 \).
2. The minimum value of \( r \) is 0 (when \( q = 0 \)), giving \( 2r = 0 \).
3. The minimum value of \( r \) is 0 (when \( q = 15 \)), giving \( 4q + 2r = 60 \).
4. The maximum value of \( r \) is 15 (when \( q = 0 \)), giving \( 2r = 30 \).

### Step 7: Determine the range of \( x \) exponents
The possible values of \( 4q + 2r \) can be expressed as:
\[ 4q + 2r = 2(2q + r) \]
We need to find the values of \( 2q + r \) where:
- \( 0 \leq q \leq 15 \)
- \( 0 \leq r \leq 15 \)

### Step 8: Calculate the distinct values
The minimum value of \( 2q + r \) occurs when \( q = 0 \) and \( r = 0 \), giving \( 0 \).
The maximum value occurs when \( q = 15 \) and \( r = 0 \) or \( q = 0 \) and \( r = 15 \), giving \( 30 \).

The values of \( 2q + r \) can take all integer values from \( 0 \) to \( 30 \), which means there are \( 31 \) distinct values.

### Conclusion
Thus, the number of dissimilar terms in the expansion of \( (1 + x^4 - 2x^2)^{15} \) is:
\[ \boxed{31} \]