This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Write out the combination terms. The given equation is ^x C_2 + ^x C_3 - 4x = 0. The formula for combinations is ^n C_r = (n!)/(r!(n-r)!). For ^x C_2: ^x C_2 = (x!)/(2!(x-2)!) = (x(x-1)(x-2)!)/(2 · 1 · (x-2)!) = (x(x-1))/(2) For ^x C_3: ^x C_3 = (x!)/(3!(x-3)!) = (x(x-1)(x-2)(x-3)!)/(3 · 2 · 1 · (x-3)!) = (x(x-1)(x-2))/(6) For these combinations to be defined, x must be an integer and x 3. Step 2: Substitute the combination terms into the equation. (x(x-1))/(2) + (x(x-1)(x-2))/(6) - 4x = 0 Step 3: Simplify the equation. Since x is a positive integer and x 3, x ≠ 0. We can divide the entire equation by x: (x-1)/(2) + ((x-1)(x-2))/(6) - 4 = 0 Multiply the entire equation by 6 to eliminate the denominators: 6((x-1)/(2)) + 6(((x-1)(x-2))/(6)) - 6(4) = 0 3(x-1) + (x-1)(x-2) - 24 = 0 Expand the terms: 3x - 3 + (x^2 - 2x - x + 2) - 24 = 0 3x - 3 + x^2 - 3x + 2 - 24 = 0 Combine like terms: x^2 + (3x - 3x) + (-3 + 2 - 24) = 0 x^2 - 25 = 0 Step 4: Solve for x. x^2 = 25 Take the square root of both sides: x = ±sqrt(25) x = ± 5 The problem states that x is a positive integer. Therefore, we choose the positive value. x = 5 This value satisfies the condition x 3. The final answer is x=5. 3 done, 2 left today. You're making progress.