Sum of the last 30 coefficients of powers of `x` in the binomial expansion of `(1 + x)^(59)` is:

Options

A. `2^(29)`
B. `2^(28)`
C. `2^(59)-2^(29)`
D. `2^(58)`

Answer

To find the sum of the last 30 coefficients of powers of \( x \) in the binomial expansion of \( (1 + x)^{59} \), we can follow these steps:

### Step 1: Understand the Binomial Expansion
The binomial expansion of \( (1 + x)^{59} \) is given by:
\[ (1 + x)^{59} = \sum_{k=0}^{59} \binom{59}{k} x^k \]
where \( \binom{59}{k} \) are the binomial coefficients.

### Step 2: Identify the Last 30 Coefficients
The last 30 coefficients correspond to the terms from \( x^{29} \) to \( x^{59} \). Thus, we need to find the sum of the coefficients:
\[ \binom{59}{29}, \binom{59}{30}, \ldots, \binom{59}{59} \]

### Step 3: Use the Property of Binomial Coefficients
We can use the property of binomial coefficients:
\[ \binom{n}{k} = \binom{n}{n-k} \]
This means:
\[ \binom{59}{29} = \binom{59}{30}, \quad \binom{59}{30} = \binom{59}{29}, \quad \ldots, \quad \binom{59}{59} = \binom{59}{0} \]
Thus, we can express the sum of the last 30 coefficients as:
\[ S = \binom{59}{29} + \binom{59}{30} + \ldots + \binom{59}{59} \]

### Step 4: Relate to the Total Sum of Coefficients
The total sum of all coefficients in the expansion is:
\[ (1 + 1)^{59} = 2^{59} \]
This includes all coefficients from \( \binom{59}{0} \) to \( \binom{59}{59} \).

### Step 5: Calculate the Sum of the First 29 Coefficients
The sum of the first 29 coefficients (from \( \binom{59}{0} \) to \( \binom{59}{28} \)) can be calculated as:
\[ \sum_{k=0}^{28} \binom{59}{k} = 2^{59} - S \]
Thus, we can express \( S \) as:
\[ S = 2^{59} - \sum_{k=0}^{28} \binom{59}{k} \]

### Step 6: Use the Symmetry of Binomial Coefficients
Using the symmetry property again, we find:
\[ \sum_{k=0}^{28} \binom{59}{k} = \sum_{k=30}^{59} \binom{59}{k} = S \]
Thus, we can conclude:
\[ S = \frac{1}{2} \cdot 2^{59} = 2^{58} \]

### Final Answer
The sum of the last 30 coefficients of powers of \( x \) in the binomial expansion of \( (1 + x)^{59} \) is:
\[ \boxed{2^{58}} \]