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 \( (1 + x^4 - 2x^2)^{15} \). We can rearrange the terms to better understand the expansion:
\[ (1 + x^4 - 2x^2)^{15} = (1 + x^4 + (-2x^2))^{15} \]

### Step 2: Identify the terms
In the expression \( (1 + x^4 - 2x^2) \), we have three distinct terms: \( 1 \), \( x^4 \), and \( -2x^2 \).

### Step 3: Use the multinomial theorem
When expanding \( (a + b + c)^n \) using the multinomial theorem, the number of distinct terms in the expansion is determined by the different combinations of the powers of \( a \), \( b \), and \( c \) that can be formed. Here, \( a = 1 \), \( b = x^4 \), and \( c = -2x^2 \).

### Step 4: Determine the powers of \( x \)
In the expansion, the general term will be of the form:
\[ \frac{15!}{k_1! k_2! k_3!} (1)^{k_1} (x^4)^{k_2} (-2x^2)^{k_3} \]
where \( k_1 + k_2 + k_3 = 15 \).

The power of \( x \) in each term will be:
\[ 4k_2 + 2k_3 \]
We need to find the distinct values of \( 4k_2 + 2k_3 \) for non-negative integers \( k_2 \) and \( k_3 \) such that \( k_1 + k_2 + k_3 = 15 \).

### Step 5: Analyze the possible values of \( k_2 \) and \( k_3 \)
Let \( k_2 \) vary from \( 0 \) to \( 15 \). For each value of \( k_2 \), \( k_3 \) can take values from \( 0 \) to \( 15 - k_2 \).

The expression \( 4k_2 + 2k_3 \) can be rewritten as:
\[ 2(2k_2 + k_3) \]
Thus, we need to analyze the term \( 2k_2 + k_3 \).

### Step 6: Determine the range of \( 2k_2 + k_3 \)
- The minimum value occurs when \( k_2 = 0 \) and \( k_3 = 0 \), giving \( 2(0) + 0 = 0 \).
- The maximum value occurs when \( k_2 = 15 \) and \( k_3 = 0 \), giving \( 2(15) + 0 = 30 \).

### Step 7: Count the distinct values
The values of \( 2k_2 + k_3 \) can range from \( 0 \) to \( 30 \). The possible values of \( k_3 \) for each \( k_2 \) will ensure that \( 2k_2 + k_3 \) takes every integer value from \( 0 \) to \( 30 \).

### Step 8: Calculate the number of dissimilar terms
Since \( 2k_2 + k_3 \) can take values from \( 0 \) to \( 30 \), the number of distinct terms in the expansion of \( (1 + x^4 - 2x^2)^{15} \) is given by:
\[ \text{Number of distinct terms} = 30 - 0 + 1 = 31 \]

### Final Answer
Thus, the number of dissimilar terms in the expansion of \( (1 + x^4 - 2x^2)^{15} \) is \( \boxed{31} \).