The total number of terms in the expansion of `(x+a)^47 - (x - a)^47` after simplification is :

Options

A. 24
B. 96
C. 47
D. 48

Answer

To find the total number of terms in the expression \((x + a)^{47} - (x - a)^{47}\) after simplification, we can follow these steps:

### Step 1: Expand both expressions using the Binomial Theorem
Using the Binomial Theorem, we can expand \((x + a)^{47}\) and \((x - a)^{47}\):

\[ (x + a)^{47} = \sum_{k=0}^{47} \binom{47}{k} x^{47-k} a^k \]

\[ (x - a)^{47} = \sum_{k=0}^{47} \binom{47}{k} x^{47-k} (-a)^k = \sum_{k=0}^{47} \binom{47}{k} x^{47-k} (-1)^k a^k \]

### Step 2: Write the difference of the two expansions
Now, we can write the difference:

\[ (x + a)^{47} - (x - a)^{47} = \sum_{k=0}^{47} \binom{47}{k} x^{47-k} a^k - \sum_{k=0}^{47} \binom{47}{k} x^{47-k} (-1)^k a^k \]

### Step 3: Combine the two expansions
Combining the two expansions, we have:

\[ = \sum_{k=0}^{47} \binom{47}{k} x^{47-k} (a^k - (-1)^k a^k) \]

This simplifies to:

\[ = \sum_{k=0}^{47} \binom{47}{k} x^{47-k} a^k (1 - (-1)^k) \]

### Step 4: Identify the terms that survive after simplification
Notice that \(1 - (-1)^k\) is zero for even \(k\) and is \(2\) for odd \(k\). Therefore, only the odd \(k\) terms will survive:

\[ = 2 \sum_{k \text{ odd}} \binom{47}{k} x^{47-k} a^k \]

### Step 5: Determine the odd values of \(k\)
The odd values of \(k\) from \(0\) to \(47\) are \(1, 3, 5, \ldots, 47\). This forms an arithmetic progression (AP) where the first term \(a = 1\), the last term \(l = 47\), and the common difference \(d = 2\).

### Step 6: Calculate the number of terms in the AP
To find the number of terms \(n\) in the AP, we can use the formula for the last term of an AP:

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

Substituting the known values:

\[ 47 = 1 + (n - 1) \cdot 2 \]

Solving for \(n\):

\[ 47 - 1 = (n - 1) \cdot 2 \implies 46 = (n - 1) \cdot 2 \implies n - 1 = 23 \implies n = 24 \]

### Conclusion
Thus, the total number of terms in the expansion of \((x + a)^{47} - (x - a)^{47}\) after simplification is \(24\).