Evaluate n+1Cn-2 If n =15

Analyze the question

We need to evaluate ⁿ⁺¹Cₙ₋₂ when n = 15.

Concept

The combination formula is:

ⁿCᵣ = n! / [r!(n − r)!]

Solution

Substitute n = 15:

• n + 1 = 16
• n − 2 = 13

So we need ¹⁶C₁₃.

¹⁶C₁₃ = 16! / (13! × 3!)

Expand the factorials so that 13! cancels:

= (16 × 15 × 14) / (3 × 2 × 1) = 3360 / 6 = 560

Check using symmetry: ¹⁶C₁₃ = ¹⁶C₃ (since ⁿCᵣ = ⁿCₙ₋ᵣ), and ¹⁶C₃ = (16 × 15 × 14)/6 = 560. ✔

Answer

The value is 560, which is option D.

Note that 3360 (option B) is the trap: it is the numerator 16 × 15 × 14, and a student who forgets to divide by 3! = 6 will pick it.

Tip

When r is close to n, use the symmetry rule ⁿCᵣ = ⁿCₙ₋ᵣ to turn it into a small calculation. Here ¹⁶C₁₃ became ¹⁶C₃, which only needs three numbers on top and 3! below.