Question

If ⁿC₉ = ⁿC₃. what is n?

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

The Correct Option is C

To solve the problem where ⁿC₉ = ⁿC₃, we use the formula for combinations, which is defined as:

C(n,k) = n! / (k! (n-k)!)

Given that ⁿC₉ = ⁿC₃, it follows that:

n! / (9! (n-9)!) = n! / (3! (n-3)!)

Cancelling n! from both sides, we have:

1 / (9! (n-9)!) = 1 / (3! (n-3)!)

By cross-multiplying, we simplify to:

3! (n-9)! = 9! (n-3)!

Because 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3!, we can rewrite this equality as:

(n-9)! = (9 × 8 × 7 × 6 × 5 × 4) × (n-3)(n-4)(n-5)(n-6)(n-7)(n-8)

Solving factorial expansions both sides lead us to compare terms directly, which simplifies to n-9 = 3, thus

n = 3 + 9 = 12.

Therefore, the value of n is 12.