Question

If ⁿC₄ = ⁿC₆, find ¹²C_n.

Answer 1

We have,
\[{}^n C_4 = {}^n C_6\]

\[\Rightarrow n = 6 + 4 = 10\]  [∵ \[{}^n C_x = {}^n C_y \Rightarrow x = y\]]  or, \[n = x + y\]

Now, \[{}^{12} C_{10} = {}^{12} C_2\][∵ \[{}^n C_r = {}^n C_{n - r}\]]

\[\Rightarrow^{12} C_{10} =^{12} C_2 = \frac{12}{2} \times \frac{11}{1} \times^{10} C_0\] [∵ \[{}^n C_r = \frac{n}{r} {}^{n - 1} C_{r - 1}\]]

\[\Rightarrow {}^{12} C_{10} = 66\]  [∵\[{}^n C_0 = 1\]]