If nC4 = nC6, find 12Cn.

1.	If nC4 = nC6, find 12Cn.
Answer» As we know that if ⁿC_p = ⁿC_q, now one of the following conditions need to be satisfied: (i) p = q (ii) n = p + q Therefore from the question ⁿC₄ = ⁿC₆, we can say that 4 ≠ 6 Therefore, the condition (ii) must be satisfied, n = 4 + 6 n = 10 Then, we need to find ¹²C_n, As we know that the value of n so, ¹²C_n = ¹²C₁₀ Lets use the formula, ⁿC_r = n!/r!(n – r)! Therefore now, value of n = 12 and r = 10 ⁿC_r = n!/r!(n – r)! ¹²C₁₀ = 12! / 10!(12 – 10)! = 12! / (10! 2!) = [12 × 11 × 10!] / (10! 2!) = [12 × 11] / (2) = 6 × 11 = 66 ∴ The value of ¹²C₁₀ = 66.

Answer»

As we know that if ⁿC_p = ⁿC_q, now one of the following conditions need to be satisfied:

(i) p = q

(ii) n = p + q

Therefore from the question ⁿC₄ = ⁿC₆, we can say that

4 ≠ 6

Therefore, the condition (ii) must be satisfied,

n = 4 + 6

n = 10

Then, we need to find ¹²C_n,

As we know that the value of n so, ¹²C_n = ¹²C₁₀

Lets use the formula,

ⁿC_r = n!/r!(n – r)!

Therefore now, value of n = 12 and r = 10

ⁿC_r = n!/r!(n – r)!

¹²C₁₀ = 12! / 10!(12 – 10)!

= 12! / (10! 2!)

= [12 × 11 × 10!] / (10! 2!)

= [12 × 11] / (2)

= 6 × 11

= 66

∴ The value of ¹²C₁₀ = 66.

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