Find the value of n if the coefficients of the sixth and sixteenth terms in the expansion of (a+b)^n are equal.

Problem 5: Find the value of n if the coefficients of the sixth and sixteenth terms in the expansion of (a+b)^n are equal. Step 1: Identify the coefficients. The coefficient of the (r+1)-th term in (a+b)^n is nr. For the sixth term, r=5, so the coefficient is n5. For the sixteenth term, r=15, so the coefficient is n15. Step 2: Set the coefficients equal and solve for n. Given n5 = n15. Using the property nr = nk r=k or r+k=n. Since 5 ≠ 15, we must have n = 5+15. n = 20 The value of n is 20. Problem 6: Expand the following: (a+b-c)^3. Step 1: Group terms and apply the binomial expansion. Let X = (a+b). Then the expression is (X-c)^3. (X-c)^3 = X^3 - 3X^2c + 3Xc^2 - c^3 Step 2: Substitute X=(a+b) back into the expansion. (a+b)^3 - 3(a+b)^2c + 3(a+b)c^2 - c^3 Step 3: Expand (a+b)^3 and (a+b)^2. (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 (a+b)^2 = a^2 + 2ab + b^2 Step 4: Substitute these expansions and simplify. (a^3 + 3a^2b + 3ab^2 + b^3) - 3(a^2 + 2ab + b^2)c + 3(a+b)c^2 - c^3 a^3 + 3a^2b + 3ab^2 + b^3 - 3a^2c - 6abc - 3b^2c + 3ac^2 + 3bc^2 - c^3 The expansion is a^3 + b^3 - c^3 + 3a^2b + 3ab^2 - 3a^2c - 3b^2c + 3ac^2 + 3bc^2 - 6abc. **Problem 7: Write down in its simplest form the complete