The total number of terms in the expansion of `(x+a)^51 - (x - a)^51` after simplification is :

Options

A. 102
B. 23
C. 26
D. none of these

Answer

To find the total number of terms in the expression \((x + a)^{51} - (x - a)^{51}\) after simplification, we can follow these steps:

### Step 1: Write the Binomial Expansions
We start by expanding both \((x + a)^{51}\) and \((x - a)^{51}\) using the Binomial Theorem.

\[ (x + a)^{51} = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} a^k \]

\[ (x - a)^{51} = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} (-a)^k = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} (-1)^k a^k \]

### Step 2: Combine the Expansions
Now, we subtract the second expansion from the first:

\[ (x + a)^{51} - (x - a)^{51} = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} a^k - \sum_{k=0}^{51} \binom{51}{k} x^{51-k} (-1)^k a^k \]

This simplifies to:

\[ = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} (a^k - (-1)^k a^k) \]

### Step 3: Simplify the Terms
Notice that \(a^k - (-1)^k a^k = a^k (1 - (-1)^k)\). This means:

- If \(k\) is even, \(1 - (-1)^k = 0\) (the term cancels out).
- If \(k\) is odd, \(1 - (-1)^k = 2\) (the term remains).

Thus, only the odd \(k\) terms will contribute to the final expression.

### Step 4: Identify the Odd Terms
The odd values of \(k\) from 0 to 51 are \(1, 3, 5, \ldots, 51\). This is an arithmetic progression (AP) where:

- First term \(a = 1\)
- Common difference \(d = 2\)
- Last term \(l = 51\)

### Step 5: Count the Number of Terms
To find the number of terms \(n\) in this AP, we can use the formula for the \(n\)-th term of an AP:

\[ l = a + (n - 1)d \]

Substituting the known values:

\[ 51 = 1 + (n - 1) \cdot 2 \]

Solving for \(n\):

\[ 51 - 1 = (n - 1) \cdot 2 \]
\[ 50 = (n - 1) \cdot 2 \]
\[ n - 1 = 25 \]
\[ n = 26 \]

### Conclusion
Thus, the total number of terms in the expansion of \((x + a)^{51} - (x - a)^{51}\) after simplification is **26**.