Question

If ⁿC₄ = ⁿC₆, find the value of ¹²C_n.

Answer 1

We know that,

If ⁿC_p = ⁿC_q,

Then one of the following conditions need to be satisfied :

i. p = q 

ii. n = p + q

From the problem,

ⁿC₄ = ⁿC₆

We can say that,

⇒ 4 ≠ 6 

So, 

The condition(ii) must be satisfied, 

⇒ n = 4 + 6 

⇒ n = 10.

We need to find the value of ¹²C_n = ¹²C₁₀

We know that,

ⁿC_r = \(\frac{n!}{r!(n-r)!}\) ..... (1)

And also,

n! = n(n – 1)(n – 2)…………2.1

From (1)

∴ The value of ¹²C₁₀ is 66.