Math with Mrs. Sarah

Generalized Binomial Coefficients

(nr)=n(n−1)(n−2)⋯(n−r+1)r!,n∈ℚ, r∈ℤ≥0\binom{n}{r} = \frac{n(n-1)(n-2)\cdots(n-r+1)}{r!}, \quad n \in \mathbb{Q},\ r \in \mathbb{Z}_{\geq 0}

Turns out, this formula is quite simple:

There are 3 cases: r = 0, r = 1, and otherwise.

If r=0, the answer is 1.

If r=1, the answer is n.

Otherwise, do these 3 steps.

Step 1: Calculate n – r + 1. This will be a factor in the numerator

Step 2: Add 1 to the value in Step 1. This will be another factor in the numerator

Step 3: Keep adding 1. Each time we add 1, we find another factor in the numerator. The last factor should be n. Then divide the numerator by r!

Now you try:

(232)\binom{\frac{2}{3}}{2}
(−51)\binom{-5}{1}
(−123)\binom{-\frac{1}{2}}{3}
(−322)\binom{-\frac{3}{2}}{2}
(−340)\binom{-\frac{3}{4}}{0}
(−33)\binom{-3}{3}
(−15)\binom{-1}{5}
(122)\binom{\frac{1}{2}}{2}
(251)\binom{\frac{2}{5}}{1}
(524)\binom{\frac{5}{2}}{4}

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