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Pascal's Formula — Definition, Formula & Examples

Pascal's Formula states that any combination (nk)\binom{n}{k} equals the sum of two smaller combinations: (n−1k−1)+(n−1k)\binom{n-1}{k-1} + \binom{n-1}{k}. It explains why each number in Pascal's Triangle is the sum of the two numbers directly above it.

For integers nn and kk with 1≤k≤n−11 \le k \le n-1, Pascal's Identity is the combinatorial identity (nk)=(n−1k−1)+(n−1k)\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}. This recurrence relation allows every binomial coefficient to be computed from two binomial coefficients of lower order.

Key Formula

(nk)=(n−1k−1)+(n−1k)\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}
Where:
  • nn = Total number of objects to choose from
  • kk = Number of objects being chosen
  • (nk)\binom{n}{k} = The number of ways to choose k objects from n (a binomial coefficient)

How It Works

Imagine you have nn objects and must choose kk of them. Focus on one specific object. Either you include it in your selection or you don't. If you include it, you still need to choose k−1k-1 more from the remaining n−1n-1 objects, giving (n−1k−1)\binom{n-1}{k-1} ways. If you exclude it, you must choose all kk from the remaining n−1n-1 objects, giving (n−1k)\binom{n-1}{k} ways. Adding these two cases gives the total (nk)\binom{n}{k}.

Worked Example

Problem: Use Pascal's Formula to verify that (62)=(51)+(52)\binom{6}{2} = \binom{5}{1} + \binom{5}{2}.
Apply the formula: Set n=6n = 6 and k=2k = 2. Pascal's Formula gives:
(62)=(51)+(52)\binom{6}{2} = \binom{5}{1} + \binom{5}{2}
Compute each term: Evaluate the three binomial coefficients using the combination formula:
(51)=5,(52)=5!2!⋅3!=10,(62)=6!2!⋅4!=15\binom{5}{1} = 5, \quad \binom{5}{2} = \frac{5!}{2! \cdot 3!} = 10, \quad \binom{6}{2} = \frac{6!}{2! \cdot 4!} = 15
Verify: Check that the right side equals the left side:
5+10=15  ✓5 + 10 = 15 \; \checkmark
Answer: The identity holds: (62)=15=5+10=(51)+(52)\binom{6}{2} = 15 = 5 + 10 = \binom{5}{1} + \binom{5}{2}.

Why It Matters

Pascal's Formula is the rule that builds Pascal's Triangle row by row, which appears throughout algebra and probability. In courses like precalculus and AP Statistics, it provides a fast way to find binomial coefficients without computing large factorials. It also underpins the proof of the Binomial Theorem by mathematical induction.

Common Mistakes

Mistake: Writing (nk)=(n−1k−1)+(n−1k+1)\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k+1} (using k+1k+1 instead of kk).
Correction: The second term keeps kk unchanged, not k+1k+1. Think of excluding the special object — you still need to choose the same number kk from fewer objects.