{{begin-eqn}} {{eqn | r = \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3       | o =        | c =  }} {{eqn | r = \left({-1}\right)^0 \dbinom 0 0 + \left({-1}\right)^1 \dbinom 1 1 + \left({-1}\right)^2 \dbinom 2 2 + \left({-1}\right)^3 \dbinom 3 3       | c = Negated Upper Index of Binomial Coefficient: Corollary 1 }} {{eqn | r = 1 - 1 + 1 - 1       | c = Binomial Coefficient with Self }} {{eqn | r = 0       | c =  }} {{eqn | r = 0^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | r = \dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3       | o =        | c =  }} {{eqn | r = 1 + 0 + 0 + 0       | c = Binomial Coefficient with Zero }} {{eqn | r = 1       | c =  }} {{eqn | r = 1^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | r = \dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3       | o =        | c =  }} {{eqn | r = 1 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3       | c = Binomial Coefficient with Zero }} {{eqn | r = 1 + 2 + \dbinom 2 2 + \dbinom 2 3       | c = Binomial Coefficient with One }} {{eqn | r = 1 + 2 + 1 + \dbinom 2 3       | c = Binomial Coefficient with Self }} {{eqn | r = 1 + 2 + 1 + 0       | c = {{Defof|Binomial Coefficient}} }} {{eqn | r = 4       | c =  }} {{eqn | r = 2^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | r = \dbinom 7 0 + \dbinom 7 1 + \dbinom 7 2 + \dbinom 7 3       | o =        | c =  }} {{eqn | r = \dfrac {7!} {7! \, 0!} + \dfrac {7!} {6! \, 1!} + \dfrac {7!} {5! \, 2!} + \dfrac {7!} {4! \, 3!}       | c = {{Defof|Binomial Coefficient}} }} {{eqn | r = \dfrac {7!} {7! \times 1} + \dfrac 7 1 + \dfrac {7 \times 6} {2 \times 1} + \dfrac {7 \times 6 \times 5} {3 \times 2 \times 1}       | c = {{Defof|Factorial}} }} {{eqn | r = 1 + 7 + \dfrac {42} 2 + \dfrac {210} 6       | c =  }} {{eqn | r = 1 + 7 + 21 + 35       | c =  }} {{eqn | r = 64       | c =  }} {{eqn | r = 8^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | r = \dbinom {15} 0 + \dbinom {15} 1 + \dbinom {15} 2 + \dbinom {15} 3       | o =        | c =  }} {{eqn | r = \dfrac {15!} {15! \, 0!} + \dfrac {15!} {14! \, 1!} + \dfrac {15!} {13! \, 2!} + \dfrac {15!} {12! \, 3!}       | c = {{Defof|Binomial Coefficient}} }} {{eqn | r = \dfrac {15!} {15! \times 1} + \dfrac {15} 1 + \dfrac {15 \times 14} {2 \times 1} + \dfrac {15 \times 14 \times 13} {3 \times 2 \times 1}       | c = {{Defof|Factorial}} }} {{eqn | r = 1 + 15 + \dfrac {210} 2 + \dfrac {2730} 6       | c =  }} {{eqn | r = 1 + 15 + 105 + 455       | c =  }} {{eqn | r = 576       | c =  }} {{eqn | r = 24^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | r = \dbinom {74} 0 + \dbinom {74} 1 + \dbinom {74} 2 + \dbinom {74} 3       | o =        | c =  }} {{eqn | r = \dfrac {74!} {74! \, 0!} + \dfrac {74!} {73! \, 1!} + \dfrac {74!} {72! \, 2!} + \dfrac {74!} {71! \, 3!}       | c = {{Defof|Binomial Coefficient}} }} {{eqn | r = \dfrac {74!} {74! \times 1} + \dfrac {74} 1 + \dfrac {74 \times 73} {2 \times 1} + \dfrac {74 \times 73 \times 72} {3 \times 2 \times 1}       | c = {{Defof|Factorial}} }} {{eqn | r = 1 + 74 + \dfrac {5402} 2 + \dfrac {388 \, 944} 6       | c =  }} {{eqn | r = 1 + 74 + 2701 + 64 \, 824       | c =  }} {{eqn | r = 67 \, 600       | c =  }} {{eqn | r = 260^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | r = \dbinom {767} 0 + \dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3       | o =        | c =  }} {{eqn | r = \dfrac {767!} {767! \, 0!} + \dfrac {767!} {766! \, 1!} + \dfrac {767!} {765! \, 2!} + \dfrac {767!} {764! \, 3!}       | c = {{Defof|Binomial Coefficient}} }} {{eqn | r = \dfrac {767!} {767! \times 1} + \dfrac {767} 1 + \dfrac {767 \times 766} {2 \times 1} + \dfrac {767 \times 766 \times 765} {3 \times 2 \times 1}       | c = {{Defof|Factorial}} }} {{eqn | r = 1 + 767 + \dfrac {587 \, 522} 2 + \dfrac {449 \, 454 \, 330} 6       | c =  }} {{eqn | r = 1 + 767 + 293 \, 761 + 74 \, 909 \, 055       | c =  }} {{eqn | r = 75 \, 203 \, 584       | c =  }} {{eqn | r = 8672^2       | c =  }} {{end-eqn}} {{ProofWanted|It remains to show there are no more.}}