Año nuevo Matemáticas$2019$

Question

$$\;\;\;\color{red}{\binom {20}{19}}\\ \color{orange}{+\binom {19}{a} +{\binom ab}}\\ \color{verde}{+\binom bc +\binom cd}\\ \color{blue}{+\binom de+ \binom ef}\\ \color{color púrpura}{+\binom fg+\binom gh}\\ \color{magenta}{+\binom hk}\\[15pt] \color{orange}=\\[10] \color{red}{2019}\\[15pt] \color{orange}{a,b} \color{verde}{c,d} \color{blue}{e,f} \color{color púrpura}{g,h} \color{magenta}k\color{orange}{=?}\\[20pt]$$ $$\color{red}{\text{Happy New Year!}}$$

---

Nota - por Favor, siéntase libre de publicar cualquier otro interesante identidades basadas en el mismo tema!

Answer 1

$$ \ color {rojo} {\ binom {20} {19}} + \ color {naranja} {\ binom {19} {16} + {\ binom {16} {14}}} + \ color {verde} {\ binom {14} {12} + \ binom {12} {7}} + \ color {azul} {\ binom {7} {6} + \ binom {6} {5}} + \ color {púrpura} {\ binom {5} {3} + \ binom {3} {1}} + \ color {magenta} {\ binom {1} {0}} = \ color {rojo} {2019} $$

Answer 2

$$\color{red}{\binom{20}{19}}+\color{orange}{\binom{19}{18}}+\color{orange}{\binom{18}{17}}+\color{green}{\binom{17}{16}}+\color{green}{\binom{16}{12}}+\color{blue}{\binom{12}{11}}+\color{blue}{\binom{11}{10}}+\color{purple}{\binom{10}{8}}+\color{purple}{\binom{8}{5}}+\color{magenta}{\binom{5}{0}}\color{orange}=\color{red}{2019}$ $ $\color{red}{\text{Happy New Year!}}$

Answer 3

Una simple secuencia de comandos de python produjo los siguientes resultados para a,b,c,d,e,f,g,h,k , de manera que $19>a> b> c> \cdots > k\geq 0$

16,14,12,7,6,5,3,1,0 16,14,12,7,6,5,3,2,0 17,14,10,7,6,5,3,1,0 17,14,10,7,6,5,3,2,0 18,15,14,10,9,5,4,2,0 18,15,14,10,9,5,4,3,1 18,15,14,10,9,5,4,3,2 18,17,16,12,10,9,7,6,1 18,17,16,12,10,9,7,6,5 18,17,16,12,11,9,7,2,0 18,17,16,12,11,9,7,5,0 18,17,16,12,11,9,7,6,2 18,17,16,12,11,9,7,6,4 18,17,16,12,11,10,8,3,0 18,17,16,12,11,10,8,5,0

Answer 4

$$ \begin{align}&{7\choose H}+\color{red}{9\choose A}+\color{blue}{11\choose P}+{\color{blue}{12\choose P}}+\color{green}{13\choose Y}+\\ &{8\choose N}+\color{magenta}{16\choose E}+{15\choose W}+\\ &\color{green}{14\choose Y}+\color{magenta}{17\choose E}+\color{red}{10\choose A}+\binom{18}{R}= 2019.\end {align} $$

$\binom76+\binom85+\binom98+\binom{10}8+\binom{11}{10}+\binom{12}{10}+\binom{13}{12}+\binom{14}{12}+\binom{15}{13}+\binom{16}{14}+\binom{17}{14}+\binom{18}{15}=2019.$

Answer 5

$\color{red}{{20 \choose 19}+{19\choose 16}}+\color{orange}{{16\choose 3}+{3\choose 1}}+\color{green}{{1\choose 2}+{2\choose 2}}+\color{blue}{{2\choose 466}+{466\choose 465}}+\color{purple}{{465\choose 466}+{466\choose 467}}=\color{magenta}{2019}$ .