The coefficient of `x^(10)` in the expansion of `(1+x)^(15)+(1+x)^(16)+(1+x)^(17)+… +(1+x)^(30)`

Options

A. `.^(31)C_(10)- ^(15)C_(10)`
B. `.^(31)C_(11)- ^(15)C_(11)`
C. `.^(30)C_(10)- ^(15)C_(10)`
D. `.^(31)C_(10)- ^(14)C_(11)`

Answer

To find the coefficient of \( x^{10} \) in the expansion of \( (1+x)^{15} + (1+x)^{16} + (1+x)^{17} + \ldots + (1+x)^{30} \), we can follow these steps:

### Step 1: Identify the series
The expression can be rewritten as:
\[ (1+x)^{15} + (1+x)^{16} + (1+x)^{17} + \ldots + (1+x)^{30} \]
This is a sum of powers of \( (1+x) \) from 15 to 30.

### Step 2: Factor out the common term
We can factor out \( (1+x)^{15} \):
\[ (1+x)^{15} \left( 1 + (1+x) + (1+x)^2 + \ldots + (1+x)^{15} \right) \]

### Step 3: Recognize the geometric series
The expression inside the parentheses is a geometric series with first term \( 1 \) and common ratio \( (1+x) \). The number of terms is \( 16 \) (from \( 0 \) to \( 15 \)). The sum of a geometric series can be calculated using the formula:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

### Step 4: Apply the formula
Here, \( a = 1 \), \( r = (1+x) \), and \( n = 16 \):
\[ S = \frac{1 \cdot ((1+x)^{16} - 1)}{(1+x) - 1} = \frac{(1+x)^{16} - 1}{x} \]

### Step 5: Substitute back into the expression
Now substituting back, we have:
\[ (1+x)^{15} \cdot \frac{(1+x)^{16} - 1}{x} \]
This simplifies to:
\[ \frac{(1+x)^{31} - (1+x)^{15}}{x} \]

### Step 6: Find the coefficient of \( x^{10} \)
To find the coefficient of \( x^{10} \) in this expression, we need to find the coefficient of \( x^{11} \) in the numerator \( (1+x)^{31} - (1+x)^{15} \) because of the division by \( x \).

Using the binomial theorem, the coefficient of \( x^k \) in \( (1+x)^n \) is given by \( \binom{n}{k} \).

Thus, we need:
\[ \text{Coefficient of } x^{11} \text{ in } (1+x)^{31} - \text{Coefficient of } x^{11} \text{ in } (1+x)^{15} \]
This gives us:
\[ \binom{31}{11} - \binom{15}{11} \]

### Final Answer
The coefficient of \( x^{10} \) in the original expression is:
\[ \binom{31}{11} - \binom{15}{11} \]