How many terms are there in the expansion of `( 1+ 2x + x^(2))^(10)` ?

Options

A. 11
B. 20
C. 21
D. 30

Answer

To find the number of terms in the expansion of \( (1 + 2x + x^2)^{10} \), we can follow these steps:

### Step 1: Identify the structure of the expression
The expression \( (1 + 2x + x^2)^{10} \) is a polynomial raised to a power. We need to analyze how many distinct terms will appear when we expand this polynomial.

### Step 2: Determine the maximum degree of the polynomial
The highest degree of the terms in the polynomial \( 1 + 2x + x^2 \) is \( 2 \) (from the term \( x^2 \)). When we raise this polynomial to the power of \( 10 \), the maximum degree of the resulting polynomial will be \( 2 \times 10 = 20 \).

### Step 3: Identify the distinct terms
To find the number of distinct terms in the expansion, we need to consider the combinations of the powers of \( 1 \), \( 2x \), and \( x^2 \) that can be formed when expanded.

Let:
- \( a \) be the number of times \( 1 \) is chosen,
- \( b \) be the number of times \( 2x \) is chosen,
- \( c \) be the number of times \( x^2 \) is chosen.

From the binomial expansion, we have:
\[ a + b + c = 10 \]

The resulting term from choosing \( a \), \( b \), and \( c \) will be of the form:
\[ x^{b + 2c} \]

### Step 4: Determine the range of powers of \( x \)
The minimum value of \( b + 2c \) occurs when \( c = 0 \) (all choices are \( 1 \) or \( 2x \)), which gives \( b \) from \( 0 \) to \( 10 \). Thus, the minimum power of \( x \) is \( 0 \) (when \( b = 0 \)).

The maximum value occurs when \( b = 0 \) (all choices are \( 1 \) or \( x^2 \)), which gives \( c \) from \( 0 \) to \( 10 \). Thus, the maximum power of \( x \) is \( 20 \) (when \( c = 10 \)).

### Step 5: Count the distinct powers of \( x \)
The powers of \( x \) can take values from \( 0 \) to \( 20 \). The distinct powers are:
\[ 0, 1, 2, \ldots, 20 \]

### Step 6: Calculate the total number of distinct terms
The total number of distinct terms is the count of integers from \( 0 \) to \( 20 \), which is \( 20 - 0 + 1 = 21 \).

Thus, the number of terms in the expansion of \( (1 + 2x + x^2)^{10} \) is **21**.

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