The number of terms in the expansion of `(1+8y+16y^(2))^(25)` is

Options

A. 49
B. 50
C. 51
D. 52

Answer

To find the number of terms in the expansion of \((1 + 8y + 16y^2)^{25}\), we can follow these steps:

### Step 1: Identify the expression
The expression we need to expand is \((1 + 8y + 16y^2)^{25}\).

### Step 2: Recognize the form of the expression
Notice that \(1 + 8y + 16y^2\) can be rewritten. We can express it as:
\[ 1 + (4y)^2 + 2 \cdot 1 \cdot (4y) \]
This resembles the expansion of \((a + b)^2\) where \(a = 1\) and \(b = 4y\).

### Step 3: Rewrite the expression
We can rewrite the expression as:
\[ (1 + 4y)^2 \]
Thus, we have:
\[ (1 + 4y)^{50} \]
because \((1 + 4y)^2\) raised to the power of 25 gives us \((1 + 4y)^{50}\).

### Step 4: Use the binomial expansion
Now, we need to find the number of terms in the expansion of \((1 + 4y)^{50}\).

### Step 5: Apply the formula for the number of terms
In the binomial expansion of \((a + b)^n\), the number of terms is given by \(n + 1\). Here, \(n = 50\).

### Step 6: Calculate the number of terms
Thus, the number of terms in the expansion of \((1 + 4y)^{50}\) is:
\[ 50 + 1 = 51 \]

### Conclusion
Therefore, the number of terms in the expansion of \((1 + 8y + 16y^2)^{25}\) is **51**.

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