Hockey-Stick Formulas for Binomial Coefficients

Formula 1

ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰! !8 <5" 8 3"< <" < <"

5œ!

8< 8

3œ<

œ œ or

Proof. The collection of size- subsets of can be partitioned into < Ö"ß ÞÞÞß 8× Ð8 < "Ñclasses where class is comprised of all subsets whose largest element is where3 3

3 œ <ß ÞÞÞß 8Þ 3 œ œThe size of class is . Hence ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰! !3" 8 3" <5"<" < <" <"

3œ<

8 8<

5œ!

after re-indexing with .5 œ 3 <

Example. ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰( $ % & '% $ $ $ $

œ

ˆ ‰74 is the number of size-4 subsets of Ö"ß #ß ÞÞÞß (×

ˆ ‰33 is the number of the size-4 subsets whose largest element is 4

ˆ ‰43 is the number of the size-4 subsets whose largest element is 5

ˆ ‰53 is the number of the size-4 subsets whose largest element is 6

ˆ ‰63 is the number of the size-4 subsets whose largest element is 7

The path below illustrates a route in Pascal's grid to node that corresponds toˆ ‰74

the subset , which is a subset that belongs to the class “subsets whose largestÖ"ß %ß &ß (×element is 7”. [Note: line segment downward-right denotes inclusion of in the set3 3 ÞÓ

Hockey Stick Formula 2

ˆ ‰ ˆ ‰ ˆ ‰! ! Š ‹8 8"3 8 84<"< <3 < 4

3œ! 4œ!

< <

œ œ or

Proof. The collection of size- subsets of can be partitioned into classes where< Ö"ß ÞÞÞß 8× Ð< "Ñclass is comprised of all subsets whose complement has largest element , where 3 W W 8 3 3w

ranges from to The size of class is since class contains the elements! <Þ 3 W − 3 3ˆ ‰83"<3

8 3 "ß 8 < 3 8 3 "..., and must have the remaining elements chosen from 1, ..., . Henceˆ ‰ ˆ ‰!8 8"3< <3

3œ!

<

œ 4 œ < 3ß. Also Re-indexing the sum with we obtain

ˆ ‰ ! !Š ‹ Š ‹8 84<" 84<"< 4 4

4œ< 4œ!

! <

œ œ Þ

Example. For and , we have 8 œ ( < œ % œˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰20

$ % & ' (" # $ % %

ˆ ‰20 is the number of the size-4 subsets such that max W ÐW Ñ œ ( % œ $w

ˆ ‰3"

wis the number of the size-4 subsets such that max W ÐW Ñ œ ( $ œ %

ˆ ‰4#

w is the number of the size-4 subsets such that max W ÐW Ñ œ ( # œ &

ˆ ‰53 is the number of the size-4 subsets such that max W ÐW Ñ œ ( " œ 'w

ˆ ‰6%

wis the number of the size-4 subsets such that max W ÐW Ñ œ ( ! œ (

The path below illustrates a route in Pascal's grid to node that corresponds toˆ ‰74

the subset , which is a subset that belongs to the class “subsets such thatÖ$ß %ß 'ß (× Wmax ” . [Note: line segment downward-right denotes inclusion of inÐW Ñ œ ( # œ & 3 3w

the setÞÓ