`""^10C_1 . ""^9C_5 + ""^10C_2. ""^9C_4 + ""^10C_3. ""^9C_3 + ""^10C_4. ""^9C_2 + ""^10C_5. ""^9C_1 + ""^10C_6 = ""^19C_6 + x ` then x =

Options

A. `-84`
B. 84
C. 81
D. `-81`

Answer

To solve the problem, we need to evaluate the expression given and find the value of \( x \) such that:

\[ \binom{10}{1} \cdot \binom{9}{5} + \binom{10}{2} \cdot \binom{9}{4} + \binom{10}{3} \cdot \binom{9}{3} + \binom{10}{4} \cdot \binom{9}{2} + \binom{10}{5} \cdot \binom{9}{1} + \binom{10}{6} = \binom{19}{6} + x \]

### Step 1: Calculate each term in the left-hand side

1. **Calculate \( \binom{10}{1} \cdot \binom{9}{5} \)**:
\[ \binom{10}{1} = 10, \quad \binom{9}{5} = \frac{9!}{5!4!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 126 \]
\[ \Rightarrow \binom{10}{1} \cdot \binom{9}{5} = 10 \cdot 126 = 1260 \]

2. **Calculate \( \binom{10}{2} \cdot \binom{9}{4} \)**:
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45, \quad \binom{9}{4} = \frac{9!}{4!5!} = 126 \]
\[ \Rightarrow \binom{10}{2} \cdot \binom{9}{4} = 45 \cdot 126 = 5670 \]

3. **Calculate \( \binom{10}{3} \cdot \binom{9}{3} \)**:
\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120, \quad \binom{9}{3} = 84 \]
\[ \Rightarrow \binom{10}{3} \cdot \binom{9}{3} = 120 \cdot 84 = 10080 \]

4. **Calculate \( \binom{10}{4} \cdot \binom{9}{2} \)**:
\[ \binom{10}{4} = 210, \quad \binom{9}{2} = 36 \]
\[ \Rightarrow \binom{10}{4} \cdot \binom{9}{2} = 210 \cdot 36 = 7560 \]

5. **Calculate \( \binom{10}{5} \cdot \binom{9}{1} \)**:
\[ \binom{10}{5} = 252, \quad \binom{9}{1} = 9 \]
\[ \Rightarrow \binom{10}{5} \cdot \binom{9}{1} = 252 \cdot 9 = 2268 \]

6. **Calculate \( \binom{10}{6} \)**:
\[ \binom{10}{6} = 210 \]

### Step 2: Sum all the calculated terms

Now, we sum all the terms calculated above:

\[ 1260 + 5670 + 10080 + 7560 + 2268 + 210 = 27148 \]

### Step 3: Calculate \( \binom{19}{6} \)

Now we calculate \( \binom{19}{6} \):
\[ \binom{19}{6} = \frac{19!}{6! \cdot 13!} = \frac{19 \times 18 \times 17 \times 16 \times 15 \times 14}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 27132 \]

### Step 4: Set up the equation to find \( x \)

Now we can set up the equation:
\[ 27148 = 27132 + x \]

### Step 5: Solve for \( x \)

Subtract \( 27132 \) from both sides:
\[ x = 27148 - 27132 = 16 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{16} \]