Question

Find $ nC_{21} $, if $ nC_{10} = nC_{12} $

• A. 1
• B. 21
• C. 22
• D. 2

Hint:

When solving for binomial coefficients, always use the symmetry property \( nC_r = nC_{n-r} \) to simplify the given equation.

Answer 1

The Correct Option is C

We are given that \( nC_{10} = nC_{12} \). From the symmetry property of binomial coefficients, we know that: \[ nC_{r} = nC_{n-r} \] So, we have: \[ nC_{10} = nC_{12} \implies 10 = n - 12 \implies n = 22 \] Thus, \( n = 22 \), and we are asked to find \( nC_{21} \), which is equal to: \[ nC_{21} = 22C_{21} \] Therefore, the answer is \( 22C_{21} \), which simplifies to 22.