The sum of the coefficient in `(x + y)^(10)`

Options

A. 1
B. 10
C. 512
D. 1024

Answer

To find the sum of the coefficients in the expression \((x + y)^{10}\), we can follow these steps:

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

### Step 2: Identify the Coefficients
In our case, we have \(n = 10\). The coefficients of the expansion are \(\binom{10}{0}, \binom{10}{1}, \binom{10}{2}, \ldots, \binom{10}{10}\).

### Step 3: Find the Sum of the Coefficients
To find the sum of the coefficients, we can substitute \(x = 1\) and \(y = 1\) into the expression:
\[ (1 + 1)^{10} \]

### Step 4: Calculate the Expression
Now, we calculate:
\[ (1 + 1)^{10} = 2^{10} \]

### Step 5: Compute the Final Value
Calculating \(2^{10}\):
\[ 2^{10} = 1024 \]

### Conclusion
Thus, the sum of the coefficients in \((x + y)^{10}\) is \(1024\).

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