Number of terms in the expansion of `(1-x)^(51) (1+x+x^(2) )^(50)` is

Options

A. 50
B. 51
C. 100
D. 102

Answer

To find the number of terms in the expansion of \((1-x)^{51} (1+x+x^2)^{50}\), we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression:
\[ (1-x)^{51} (1+x+x^2)^{50} \]

### Step 2: Expand \((1+x+x^2)^{50}\)
We can rewrite \(1+x+x^2\) as:
\[ 1+x+x^2 = \frac{1-x^3}{1-x} \]
Thus, we can express \((1+x+x^2)^{50}\) as:
\[ (1+x+x^2)^{50} = \left(\frac{1-x^3}{1-x}\right)^{50} = (1-x^3)^{50} (1-x)^{-50} \]

### Step 3: Combine the Expressions
Now substituting back into our original expression:
\[ (1-x)^{51} \cdot (1-x^3)^{50} \cdot (1-x)^{-50} \]
This simplifies to:
\[ (1-x)^{51-50} \cdot (1-x^3)^{50} = (1-x)^{1} \cdot (1-x^3)^{50} \]
So we have:
\[ (1-x)(1-x^3)^{50} \]

### Step 4: Expand \((1-x^3)^{50}\)
Using the binomial theorem, the expansion of \((1-x^3)^{50}\) will have terms of the form:
\[ \binom{50}{k} (-x^3)^k = \binom{50}{k} (-1)^k x^{3k} \]
where \(k\) can take values from \(0\) to \(50\).

### Step 5: Combine the Terms
Now, we need to consider the product:
\[ (1-x)(1-x^3)^{50} \]
This will yield:
\[ 1 \cdot (1-x^3)^{50} - x \cdot (1-x^3)^{50} \]
The first term contributes the same number of terms as \((1-x^3)^{50}\), which has \(50 + 1 = 51\) terms.

The second term, \(-x(1-x^3)^{50}\), will also contribute \(51\) terms, but they will be of the form \(x^{3k+1}\) for \(k = 0, 1, 2, \ldots, 50\).

### Step 6: Determine Unique Terms
Now we need to find the unique terms from both expansions:
- The first expansion contributes terms of the form \(x^{3k}\) for \(k = 0, 1, 2, \ldots, 50\) (51 terms).
- The second expansion contributes terms of the form \(x^{3k+1}\) for \(k = 0, 1, 2, \ldots, 50\) (51 terms).

### Step 7: Count Total Unique Terms
The powers of \(x\) from both expansions are:
- From the first: \(0, 3, 6, \ldots, 150\) (which gives us 51 terms).
- From the second: \(1, 4, 7, \ldots, 151\) (which also gives us 51 terms).

Since the powers \(0, 3, 6, \ldots, 150\) and \(1, 4, 7, \ldots, 151\) do not overlap, the total number of unique terms is:
\[ 51 + 51 = 102 \]

### Final Answer
Thus, the number of terms in the expansion of \((1-x)^{51} (1+x+x^2)^{50}\) is:
\[ \boxed{102} \]