Find ^n C₂₁ , if ^n C₁₀= ^n C₁₂

Step-by-step solution

The property of combinations states that if ^ n C_ a = ^ n C_ b , then either a = b or a + b = n . Given ^ n C₁₀ = ^ n C₁₂ , since 10 12 , it must be that 10 + 12 = n . Therefore, n = 22 . We are asked to find ^ n C₂₁ , which is ²²C₂₁ . Using the property ^ n C_ r = ^ n C_ n-r , we have ²²C₂₁ = ²²C_ 22-21 = ²²C₁ . Since ^ n C₁ = n , we get ²²C₁ = 22 . Answer: 22