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 General Term
The general term in the expansion of \( (1+x)^n \) is given by:
\[ T_k = \binom{n}{k} x^k \]
where \( n \) is the exponent and \( k \) is the term number.

### Step 2: Find the Coefficient of \( x^{10} \)
We need to find the coefficient of \( x^{10} \) for each \( n \) from 15 to 30. Thus, we need to calculate:
\[ \binom{15}{10} + \binom{16}{10} + \binom{17}{10} + \ldots + \binom{30}{10} \]

### Step 3: Calculate Each Binomial Coefficient
We will calculate each binomial coefficient:
- For \( n = 15 \):
\[ \binom{15}{10} = \binom{15}{5} = 3003 \]
- For \( n = 16 \):
\[ \binom{16}{10} = \binom{16}{6} = 8008 \]
- For \( n = 17 \):
\[ \binom{17}{10} = \binom{17}{7} = 19448 \]
- For \( n = 18 \):
\[ \binom{18}{10} = \binom{18}{8} = 43758 \]
- For \( n = 19 \):
\[ \binom{19}{10} = \binom{19}{9} = 92378 \]
- For \( n = 20 \):
\[ \binom{20}{10} = 184756 \]
- For \( n = 21 \):
\[ \binom{21}{10} = 352716 \]
- For \( n = 22 \):
\[ \binom{22}{10} = 646646 \]
- For \( n = 23 \):
\[ \binom{23}{10} = 1144066 \]
- For \( n = 24 \):
\[ \binom{24}{10} = 1961256 \]
- For \( n = 25 \):
\[ \binom{25}{10} = 3268760 \]
- For \( n = 26 \):
\[ \binom{26}{10} = 5311735 \]
- For \( n = 27 \):
\[ \binom{27}{10} = 8436285 \]
- For \( n = 28 \):
\[ \binom{28}{10} = 1307504 \]
- For \( n = 29 \):
\[ \binom{29}{10} = 20030010 \]
- For \( n = 30 \):
\[ \binom{30}{10} = 30045015 \]

### Step 4: Sum the Coefficients
Now, we sum all these coefficients:
\[ 3003 + 8008 + 19448 + 43758 + 92378 + 184756 + 352716 + 646646 + 1144066 + 1961256 + 3268760 + 5311735 + 8436285 + 1307504 + 20030010 + 30045015 \]

### Step 5: Final Calculation
Calculating the total gives:
\[ 3003 + 8008 + 19448 + 43758 + 92378 + 184756 + 352716 + 646646 + 1144066 + 1961256 + 3268760 + 5311735 + 8436285 + 1307504 + 20030010 + 30045015 = 30045015 \]

### Final Answer
The coefficient of \( x^{10} \) in the given expansion is:
\[ \boxed{30045015} \]