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Question Number 185642 by Mingma last updated on 24/Jan/23

Commented by mr W last updated on 24/Jan/23

=C_(4+1) ^(26+1) =C_5 ^(27) =80730

=C4+126+1=C527=80730

Answered by cortano1 last updated on 25/Jan/23

Σ_(k=4) ^n C_4 ^k = (1/(4!))((((n−3)(n−2)(n−1)n(n+1))/5))

∑nk=4C4k=14!((n−3)(n−2)(n−1)n(n+1)5)

Commented by mr W last updated on 25/Jan/23

Σ_(k=4) ^n C_4 ^k = (1/(4!))((((n−3)(n−2)(n−1)n(n+1))/5)) =(((n+1)!)/(5!(n−4)!))=C_5 ^(n+1) generally Σ_(k=r) ^n ((k),(r) ) = (((n+1)),((r+1)) ) or Σ_(k=r) ^n C_r ^k =C_(r+1) ^(n+1) this is the Hockey−stick identity.

∑nk=4C4k=14!((n−3)(n−2)(n−1)n(n+1)5)=(n+1)!5!(n−4)!=C5n+1generally∑nk=r(kr)=(n+1r+1)or∑nk=rCrk=Cr+1n+1thisistheHockey−stickidentity.

Commented by mr W last updated on 25/Jan/23

proof see Q185659

proofseeQ185659

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