Write the given ratio in fractional form and simplify the right side.

Alright agyenim — let's do this. To find the value of n from the given ratio, we will use the combination formula ^kC_r = (k!)/(r!(k-r)!). Step 1: Write the given ratio in fractional form and simplify the right side. (^8C_n)/(^6C_n-1) = (56)/(16) Simplify the ratio on the right side by dividing both numbers by their greatest common divisor, which is 8: (^8C_n)/(^6C_n-1) = (7)/(2) Step 2: Expand the combination terms using the formula. ^8C_n = (8!)/(n!(8-n)!) ^6C_n-1 = (6!)/((n-1)!(6-(n-1))!) = (6!)/((n-1)!(7-n)!) Step 3: Substitute the expanded forms into the ratio and simplify. (8!)/(n!(8-n)!)(6!)/((n-1)!(7-n)!) = (7)/(2) Invert and multiply: (8!)/(n!(8-n)!) × ((n-1)!(7-n)!)/(6!) = (7)/(2) Expand the factorials to cancel common terms: 8! = 8 × 7 × 6! n! = n × (n-1)! (8-n)! = (8-n) × (7-n)! Substitute these into the equation: (8 × 7 × 6!)/(n × (n-1)! × (8-n) × (7-n)!) × ((n-1)!(7-n)!)/(6!) = (7)/(2) Cancel out 6!, (n-1)!, and (7-n)!: (8 × 7)/(n(8-n)) = (7)/(2) (56)/(n(8-n)) = (7)/(2) Step 4: Solve the equation for n. Cross-multiply: 56 × 2 = 7 × n(8-n) 112 = 7(8n - n^2) Divide both sides by 7: (112)/(7) = 8n - n^2 16 = 8n - n^2 Rearrange into a standard quadratic equation: n^2 - 8n + 16 = 0 This is a perfect square trinomial: (n-4)^2 = 0 Take the square root of both sides: n-4 = 0 n = 4 Step 5: Verify the value of n. For ^8C_n, n must be between 0 and 8 (inclusive). n=4 satisfies this. For ^6C_n-1, n-1 must be between 0 and 6 (inclusive). If n=4, then n-1=3, which satisfies 0 3 6. Both conditions are met. The value of n is 4. Two more to go! Send the next one 📸