The coefficient of `x^(6)` in the expansion of `(1+x)^(21) +(1+x)^(22) + …+ (1+x)^(30)` is

Options

A. `""^(51)C_(6)`
B. `""^(9)C_(6)`
C. `""^(31)C_(7)-""^(21)C_(7)`
D. `""^(30)C_(6)+""^(20)C_(6)`

Answer

To find the coefficient of \( x^6 \) in the expansion of \( (1+x)^{21} + (1+x)^{22} + \ldots + (1+x)^{30} \), we can follow these steps:

### Step 1: Identify the series
The expression can be rewritten as:
\[ S = (1+x)^{21} + (1+x)^{22} + (1+x)^{23} + (1+x)^{24} + (1+x)^{25} + (1+x)^{26} + (1+x)^{27} + (1+x)^{28} + (1+x)^{29} + (1+x)^{30} \]
This is the sum of 10 terms, where the powers of \( (1+x) \) range from 21 to 30.

### Step 2: Use the formula for the coefficient
The coefficient of \( x^r \) in the expansion of \( (1+x)^n \) is given by \( \binom{n}{r} \). Therefore, we need to find the coefficients of \( x^6 \) in each of the expansions.

### Step 3: Calculate the coefficients
We need to find the coefficient of \( x^6 \) in each term:
- For \( (1+x)^{21} \), the coefficient of \( x^6 \) is \( \binom{21}{6} \).
- For \( (1+x)^{22} \), the coefficient of \( x^6 \) is \( \binom{22}{6} \).
- For \( (1+x)^{23} \), the coefficient of \( x^6 \) is \( \binom{23}{6} \).
- For \( (1+x)^{24} \), the coefficient of \( x^6 \) is \( \binom{24}{6} \).
- For \( (1+x)^{25} \), the coefficient of \( x^6 \) is \( \binom{25}{6} \).
- For \( (1+x)^{26} \), the coefficient of \( x^6 \) is \( \binom{26}{6} \).
- For \( (1+x)^{27} \), the coefficient of \( x^6 \) is \( \binom{27}{6} \).
- For \( (1+x)^{28} \), the coefficient of \( x^6 \) is \( \binom{28}{6} \).
- For \( (1+x)^{29} \), the coefficient of \( x^6 \) is \( \binom{29}{6} \).
- For \( (1+x)^{30} \), the coefficient of \( x^6 \) is \( \binom{30}{6} \).

### Step 4: Sum the coefficients
Now, we sum these coefficients:
\[ \text{Total coefficient} = \binom{21}{6} + \binom{22}{6} + \binom{23}{6} + \binom{24}{6} + \binom{25}{6} + \binom{26}{6} + \binom{27}{6} + \binom{28}{6} + \binom{29}{6} + \binom{30}{6} \]

### Step 5: Use the Hockey Stick Identity
According to the Hockey Stick Identity in combinatorics:
\[ \sum_{k=r}^{n} \binom{k}{r} = \binom{n+1}{r+1} \]
In our case, we can apply this identity:
\[ \sum_{k=21}^{30} \binom{k}{6} = \binom{31}{7} \]

### Final Step: Calculate the value
Thus, the coefficient of \( x^6 \) in the given expansion is:
\[ \binom{31}{7} \]