Question

Va roooooog daca puteti sa ma ajutati si pe mine la aceste exercitii......va roooooooooog.☺☺☺☺

Answer 1

$\displaystyle\mathtt{13.~~~~C_8^5-C_8^3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathbf{C_n^k= \frac{n!}{(n-k)!\cdot k!}}}\\\\\mathtt{C_8^5=\frac{8!}{(8-5)!\cdot 5!}=\frac{8!}{3! \cdot 5!}=\frac{5!\cdot6\cdot7\cdot 8}{3!\cdot5!}=\frac{6\cdot7\cdot 8}{1\cdot2\cdot 3}=7\cdot 8=56}\\\\\mathtt{C_8^5=56}\\\\\mathtt{C_8^3= \frac{8!}{(8-3)!\cdot3!}=\frac{8!}{5!\cdot3!}=\frac{5!\cdot6\cdot7\cdot8}{5! \cdot3!}=\frac{6\cdot7\cdot8}{1\cdot2\cdot 3}=7\cdot 8=56}\\\\\mathtt{C_8^3=56}\\\\\mathtt{C_8^5-C_8^3=56-56=0}$

$\displaystyle \mathtt{14.~~~~C_5^1+C_5^3+C_5^5=2^4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2^4=16}\\ \\ \mathtt{\mathbf{C_n^1=n}\Rightarrow C_5^1=5}\\ \\ \mathtt{\mathbf{C_n^k= \frac{n!}{(n-k)! \cdot k!} }\Rightarrow C_5^3= \frac{5!}{(5-3)!\cdot 3!}= \frac{5!}{2! \cdot 3!}= \frac{3! \cdot 4 \cdot 5}{2! \cdot 3!}= \frac{4 \cdot 5}{1 \cdot 2}=10 }\\ \\ \mathtt{\mathbf{C_n^n=1 }\Rightarrow C_5^5=1}\\ \\ \mathtt{C_5^1+C_5^3+C_5^5=5+10+1=16=2^4}$

$\displaystyle \mathtt{15.~~~~C_5^1+1=3!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3!=1 \cdot 2 \cdot 3=6}\\ \\ \mathtt{\mathbf{C_n^1=n }\Rightarrow C_5^1=5}\\ \\ \mathtt{C_5^1+1=5+1=6=3!}$

$\displaystyle \mathtt{16.~~~~0!+1!+2!+3!=1+1+1 \cdot 2+1 \cdot 2 \cdot 3=1+1+2+6=10}\\ \\ \mathtt{0!+1!+2!+3!=10}$