⁹C₄ — Combination Value and Formula

⁹C₄ = 126. This is calculated using the combinations formula: ⁿCᵣ = n! / (r! × (n−r)!)

Question (Click to Flip)

What is ⁹P₄ (permutation)?

Answer

⁹P₄ = 9!/(9-4)! = 9!/5! = 9×8×7×6 = 3,024. This is 4! times ⁹C₄: 24 × 126 = 3024 ✓

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Key Facts

Pascal's Triangle contains all combination values! The rth element in the nth row of Pascal's Triangle equals ⁿCᵣ. The sum of the nth row = 2ⁿ (total subsets).

Formula and Calculation

Combinations Formula: ⁿCᵣ = n! / [r! × (n−r)!]

For ⁹C₄:

n = 9, r = 4
⁹C₄ = 9! / (4! × 5!)
= (9 × 8 × 7 × 6) / (4 × 3 × 2 × 1)
= 3024 / 24
= 126

Step-by-Step

Numerator: 9 × 8 × 7 × 6 = 3024 Denominator: 4! = 4 × 3 × 2 × 1 = 24

3024 ÷ 24 = 126

⁹C₄ = 126

Verification using symmetry: ⁹C₄ = ⁹C₅ (since 9−4=5) ⁹C₅ = 9!/(5!×4!) = same calculation = 126 ✓

What Does ⁹C₄ Mean?

⁹C₄ represents the number of ways to choose 4 items from 9 distinct items where order does NOT matter.

Example: From a class of 9 students, how many ways can you select a committee of 4? Answer: ⁹C₄ = 126 ways

Key difference:

Combination (ⁿCᵣ): Order doesn't matter — {A,B,C,D} = {D,C,B,A}
Permutation (ⁿPᵣ): Order matters — {A,B,C,D} ≠ {D,C,B,A}

Questions and Answers

What is ⁹P₄ (permutation)?+

⁹P₄ = 9!/(9-4)! = 9!/5! = 9×8×7×6 = **3,024**. This is 4! times ⁹C₄: 24 × 126 = 3024 ✓

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