Question

Aranjamente de 4 luate cate 4 + Combinari de 4 luate cate 4 = ???
stiu ca Combinari de 4 luate cate 4 = 1 dar la aranjamente nu cred ca e la fel, ajutor va rog!!

Answer 1

Răspuns:

$\displaystyle C_4^4+A_4^4=24+1=25$

Explicație pas cu pas:

Folosim formulele:

$\displaystyle \boxed{C_n^k= \[ {n! \over k!(n-k)!} } \ - \text{combinari de n luate cate k}$

$\displaystyle \boxed{A_n^k= \[ {n! \over (n-k)!} } \ - \text{aranjamente de n luate cate k}$

$n! = 1 \cdot 2 \cdot 3 \cdot ... \cdot n \ - \text{factorial}$

Si acum sa calculam

$\displaystyle C_4^4=\[ {4! \over 4!(4-4)!} = \frac{1\cdot2\cdot3\cdot4}{1\cdot2\cdot3\cdot4\cdot0!}=\frac{1\cdot2\cdot3\cdot4}{1\cdot2\cdot3\cdot4\cdot1}  =1 \\ \\ \text{24 supra 24 se simplica si ramane 1}$

Retinem ca 0!=1

$\displaystyle A_4^4= \frac{4!}{(4-4)!}=\frac{4!}{0!}=1\cdot2\cdot3\cdot4=24$

Asadar, 24+1=25