C_(4)^2 cdot C_(x+1)^x-2=2 C_(x)^2-1-C_(x+3)^x+2+2 C_(7)^3

To solve the given equation, we need to simplify and manipulate the combinatorial expressions. Let's start by writing down the equation:<br /><br />\[ C_{4}^{2} \cdot C_{x+1}^{x-2} = 2 C_{x}^{2} - 1 - C_{x+3}^{x+2} + 2 C_{7}^{3} \]<br /><br />First, let's simplify the left-hand side:<br /><br />\[ C_{4}^{2} = \frac{4!}{2!(4-2)!} = \frac{4 \cdot 3}{2 \cdot 1} = 6 \]<br /><br />So the left-hand side becomes:<br /><br />\[ 6 \cdot C_{x+1}^{x-2} \]<br /><br />Now let's simplify the right-hand side:<br /><br />\[ 2 C_{x}^{2} - 1 - C_{x+3}^{x+2} + 2 C_{7}^{3} \]<br /><br />We know that:<br /><br />\[ C_{x+3}^{x+2} = C_{x+3}^{1} = x+3 \]<br /><br />And:<br /><br />\[ C_{7}^{3} = \frac{7!}{3!(7-3)!} = \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1} = 35 \]<br /><br />So the right-hand side becomes:<br /><br />\[ 2 C_{x}^{2} - 1 - (x+3) + 2 \cdot 35 \]<br /><br />\[ = 2 C_{x}^{2} - 1 - x - 3 + 70 \]<br /><br />\[ = 2 C_{x}^{2} - x + 66 \]<br /><br />Now we have:<br /><br />\[ 6 \cdot C_{x+1}^{x-2} = 2 C_{x}^{2} - x + 66 \]<br /><br />Next, we need to express \( C_{x+1}^{x-2} \) in terms of \( C_{x}^{2} \):<br /><br />\[ C_{x+1}^{x-2} = \frac{(x+1)!}{(x-2)! \cdot (x+1- (x-2))!} = \frac{(x+1)!}{(x-2)! \cdot 3!} = \frac{(x+1) \cdot x \cdot (x-1) \cdot (x-2)}{6} \cdot C_{x}^{2} \]<br /><br />So:<br /><br />\[ 6 \cdot \frac{(x+1) \cdot x \cdot (x-1) \cdot (x-2)}{6} \cdot C_{x}^{2} = 2 C_{x}^{2} - x + 66 \]<br /><br />\[ (x+1) \cdot x \cdot (x-1) \cdot (x-2) \cdot C_{x}^{2} = 2 C_{x}^{2} - x + 66 \]<br /><br />\[ (x^2 - 1)(x^2 - 2x) \cdot C_{x}^{2} = 2 C_{x}^{2} - x + 66 \]<br /><br />\[ (x^4 - 2x^3 - x^2 + 2x) \cdot C_{x}^{2} = 2 C_{x}^{2} - x + 66 \]<br /><br />\[ (x^4 - 2x^3 - x^2 + 2x - 2) \cdot C_{x}^{2} = -x + 66 \]<br /><br />This equation is quite complex to solve algebraically. It might be more practical to test specific values of \( x \) to see if they satisfy the equation. For example, if \( x = 3 \):<br /><br />\[ (3^4 - 2 \cdot 3^3 - 3^2 + 2 \cdot 3 - 2) \cdot C_{3}^{2} = -3 + 66 \]<br /><br />\[ (81 - 54 - 9 + 6 - 2) \cdot 3 = 63 \]<br /><br />\[ 22 \cdot 3 = 63 \]<br /><br />This does not hold, so we need to find the correct value of \( x \). This suggests that the equation might not have simple integer solutions and may require numerical methods or further algebraic manipulation to solve.