Il coefficiente binomiale è definito da
(dove n! è il fattoriale di n) e può essere calcolato anche facendo ricorso al triangolo di Tartaglia.
e ha le seguenti proprietà:
( n 0 ) = n ! 0 ! ( n − 0 ) ! = n ! n ! = 1 {\displaystyle {n \choose 0}={{n!} \over {0!(n-0)!}}={n! \over n!}=1}
( n k + 1 ) + ( n k ) = n ! ( k + 1 ) ! ( n − k − 1 ) ! + n ! k ! ( n − k ) ! {\displaystyle {n \choose k+1}+{n \choose k}={{n!} \over {(k+1)!(n-k-1)!}}+{{n!} \over {k!(n-k)!}}}
considerando il fatto che ( n − k ) ! = ( n − k ) ( n − k − 1 ) ! {\displaystyle (n-k)!=(n-k)(n-k-1)!} , ed allo stesso modo ( k + 1 ) ! = ( k + 1 ) k ! {\displaystyle (k+1)!=(k+1)k!} si ha ( n k + 1 ) + ( n k ) = n ! ( k + 1 ) k ! ( n − k − 1 ) ! + n ! ( n − k ) k ! ( n − k − 1 ) ! {\displaystyle {n \choose k+1}+{n \choose k}={{n!} \over {(k+1)k!(n-k-1)!}}+{{n!} \over {(n-k)k!(n-k-1)!}}}
da cui si ottiene
( n − k ) n ! ( k + 1 ) ( n − k ) k ! ( n − k − 1 ) ! + ( k + 1 ) n ! ( k + 1 ) ( n − k ) k ! ( n − k − 1 ) ! {\displaystyle {(n-k){n!} \over {(k+1)(n-k)k!(n-k-1)!}}+{(k+1){n!} \over {(k+1)(n-k)k!(n-k-1)!}}}
e quindi
( n k + 1 ) + ( n k ) = ( n − k + k + 1 ) n ! ( k + 1 ) k ! ( n − k ) ( n − k − 1 ) ! {\displaystyle {n \choose k+1}+{n \choose k}={(n-k+k+1){n!} \over {(k+1)k!(n-k)(n-k-1)!}}}
( n k + 1 ) + ( n k ) = ( n + 1 ) ! ( k + 1 ) ! ( n − k ) ! = ( n + 1 k + 1 ) {\displaystyle {n \choose k+1}+{n \choose k}={{(n+1)!} \over {(k+1)!(n-k)!}}={n+1 \choose k+1}}
, ed allo stesso modo 
si ha
![{\displaystyle {n \choose 1}={{n!} \over {1!(n-1)!}}={{n!} \over {(n-1)![n-(n-1)]!}}={n \choose n-1}=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca388767eab2d834da93a0471db8081f675aaaf1)
![{\displaystyle {n \choose k}={{n!} \over {k!(n-k)!}}={{n!} \over {(n-k)![n-(n-k)]!}}={n \choose n-k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dbf2fa7306b111f74a3bd8229f04394fc2b1139)
