Hockey stick & Vandermonde identities
The hockey stick identity Σ_{i=r}^{n} C(i,r) = C(n+1, r+1) sums a diagonal of Pascal’s triangle, and the picture of that diagonal plus its answer is what gives it the name. Vandermonde’s identity Σ_k C(m,k)C(n,r−k) = C(m+n, r) counts an r-subset of a combined set by how it splits — both have clean combinatorial proofs and neither needs algebra.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Hockey stick — count by the largest elementThe (r+1)-subsets of {0,…,n} classified by their maximum element i give exactly C(i,r) each.
- Vandermonde — split the setChoosing r from m + n objects means choosing k from the first group and r − k from the second, summed over k.
- Check the index rangesTerms with k > m or r − k > n are zero, since C(a,b) = 0 when b > a. This lets the sum range be sloppy without error.
- Verify on a small caseTest with m = n = 2, r = 2 to confirm the identity before using it on the actual problem.
Worked example
Verify Vandermonde’s identity for m = 3, n = 2, r = 2.
- Right side: C(5,2) = 10.
- Left side: Σ_k C(3,k)C(2,2−k) for k = 0, 1, 2.
- k=0: 1×1 = 1. k=1: 3×2 = 6. k=2: 3×1 = 3.
- Sum: 1 + 6 + 3 = 10.
Answer. Both sides equal 10, confirming the identity for these values.
Where marks get dropped
These are the specific errors that cost credit on hockey stick & vandermonde identities questions — QED's rubric penalises each of them separately.
- Getting the hockey stick’s upper index wrong. It is C(n+1, r+1) — both indices increase by one.
- Summing Vandermonde over the wrong range and missing terms. Extend the range freely; out-of-range binomials are zero.
- Trying to prove these algebraically. The combinatorial arguments are two lines each; the algebra is much longer.
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Hockey stick & Vandermonde identities — frequently asked questions
Why is it called the hockey stick?
Because on Pascal’s triangle the summed diagonal plus the answer traces the shape of a hockey stick — the shaft along the diagonal and the blade at the result.
What is Vandermonde with m = n = r?
It gives ΣC(n,k)² = C(2n,n), a much-used identity — count how an n-subset of a 2n-set splits between the two halves.
Where does hockey stick get used?
Summing triangular or tetrahedral numbers, and counting monotone lattice paths — anywhere a running total of binomials appears.
The rest of Combinatorics
Counting principles, permutations, combinations. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Sum & product rules
- 2Permutations & combinations
- 3Binomial theorem & Pascal’s triangle
- 4The pigeonhole principle
- 5Inclusion–exclusion
- 6Counting with repetition (stars & bars)
- 7Derangements & counting surjections
- 8Double counting & bijective proofs
- 9Setting up & solving counting recurrences
- 10Generating functions — an introduction
- 11Multinomial coefficients & repeated items
- 12Hockey stick & Vandermonde identities
- 13Catalan numbers & lattice-path counting
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