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: Determine the formula for the coefficient of a term in a binomial expansion. The (r+1)-th term in the expansion of (a+b)^n is given by T_r+1 = nr a^n-r b^r. The coefficient of the (r+1)-th term is nr. Step 2: Find the coefficients of the sixth and sixteenth terms. For the sixth term, r+1=6, so r=5. The coefficient is n5. For the sixteenth term, r+1=16, so r=15. The coefficient is n15. Step 3: Set the coefficients equal and solve for n. Given that the coefficients are equal: n5 = n15 According to the property of binomial coefficients, if nr = nk, then either r=k or r+k=n. Since 5 ≠ 15, we must have 5+15=n. n = 5+15 n = 20 The value of n is: 20 Step 1: Group the terms to apply the binomial theorem. Let X = (a+b). The expression becomes (X-c)^3. Step 2: Expand (X-c)^3 using the binomial theorem. The binomial theorem states (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3. Here, A=X and B=-c. (X-c)^3 = X^3 + 3X^2(-c) + 3X(-c)^2 + (-c)^3 = X^3 - 3cX^2 + 3c^2X - c^3 Step 3: Substitute X=(a+b) back into the expanded expression. (a+b-c)^3 = (