BINOMNI KOEFICIJENTI
BINOMNI KOEFICIJENTI
BINOMNI KOEFICIJENTI
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<strong>BINOMNI</strong> <strong>KOEFICIJENTI</strong><br />
Korolar 1<br />
( ) ( )<br />
k k + 1<br />
+<br />
k k<br />
( ) (<br />
n ∑n−k<br />
+ . . . + =<br />
k<br />
j=0<br />
k + j<br />
k<br />
) ( )<br />
n + 1<br />
= .<br />
k + 1<br />
( ) ( )<br />
k + j k + j<br />
Dokaz: Teorem 1 povlači = pa suma na desnoj strani postaje<br />
k j<br />
( ) ( ) ( ) ( )<br />
k k + 1 k + (n − k) ∑n−k<br />
k + j<br />
+ + . . . +<br />
= .<br />
0 1<br />
n − k<br />
j<br />
Ovo je suma iz Teorema 5 uz zamjenu k ↔ n, r ↔ n − k, pa je onda ona jednaka<br />
(<br />
) ( ) ( )<br />
k + (n − k) + 1 n + 1 n + 1<br />
= = .<br />
n − k n − k k + 1<br />
j=0<br />
() 21. studenog 2011. 12 / 27