Coefficient of `x^7` in `(1 + x)^10 + x(1 + x)^9 + x^2 (1 + x)^8+………..+x^10` is

Options

A. 210
B. 330
C. 420
D. 120

Answer

To find the coefficient of \( x^7 \) in the expression

\[ (1 + x)^{10} + x(1 + x)^9 + x^2(1 + x)^8 + \ldots + x^{10}, \]

we can start by rewriting the expression in a more manageable form.

### Step 1: Rewrite the Expression
The given expression can be rewritten as:

\[ \sum_{k=0}^{10} x^k (1 + x)^{10 - k}. \]

This means we are summing terms where \( k \) varies from 0 to 10, and for each \( k \), we have \( x^k \) multiplied by \( (1 + x)^{10 - k} \).

### Step 2: Expand Each Term
Next, we need to expand \( (1 + x)^{10 - k} \) using the binomial theorem:

\[ (1 + x)^{10 - k} = \sum_{j=0}^{10-k} \binom{10-k}{j} x^j. \]

Thus, the term \( x^k (1 + x)^{10 - k} \) becomes:

\[ x^k \sum_{j=0}^{10-k} \binom{10-k}{j} x^j = \sum_{j=0}^{10-k} \binom{10-k}{j} x^{k+j}. \]

### Step 3: Collect Terms
Now, we want to find the coefficient of \( x^7 \) in the entire sum. This means we need to find all pairs \( (k, j) \) such that \( k + j = 7 \).

### Step 4: Determine Valid \( k \) and \( j \)
For \( k + j = 7 \):

- If \( k = 0 \), then \( j = 7 \) (from \( (1 + x)^{10} \)).
- If \( k = 1 \), then \( j = 6 \) (from \( x(1 + x)^9 \)).
- If \( k = 2 \), then \( j = 5 \) (from \( x^2(1 + x)^8 \)).
- If \( k = 3 \), then \( j = 4 \) (from \( x^3(1 + x)^7 \)).
- If \( k = 4 \), then \( j = 3 \) (from \( x^4(1 + x)^6 \)).
- If \( k = 5 \), then \( j = 2 \) (from \( x^5(1 + x)^5 \)).
- If \( k = 6 \), then \( j = 1 \) (from \( x^6(1 + x)^4 \)).
- If \( k = 7 \), then \( j = 0 \) (from \( x^7(1 + x)^3 \)).

### Step 5: Calculate Coefficients
Now we can find the coefficients for each of these cases:

1. For \( k = 0, j = 7 \): Coefficient = \( \binom{10}{7} = 120 \).
2. For \( k = 1, j = 6 \): Coefficient = \( \binom{9}{6} = 84 \).
3. For \( k = 2, j = 5 \): Coefficient = \( \binom{8}{5} = 56 \).
4. For \( k = 3, j = 4 \): Coefficient = \( \binom{7}{4} = 35 \).
5. For \( k = 4, j = 3 \): Coefficient = \( \binom{6}{3} = 20 \).
6. For \( k = 5, j = 2 \): Coefficient = \( \binom{5}{2} = 10 \).
7. For \( k = 6, j = 1 \): Coefficient = \( \binom{4}{1} = 4 \).
8. For \( k = 7, j = 0 \): Coefficient = \( \binom{3}{0} = 1 \).

### Step 6: Sum All Coefficients
Now we sum all these coefficients:

\[ 120 + 84 + 56 + 35 + 20 + 10 + 4 + 1 = 330. \]

### Final Answer
Thus, the coefficient of \( x^7 \) in the given expression is

\[ \boxed{330}. \]