Представим эту периодическую дробь в виде суммы:
5,(05)=5+(0,05+0,0005+0,000005+...)
Найдем параметры этой бесконечной геометрической прогрессии:
x1=0,05, x2=0,0005
q=x2:x1=0,0005:0,05=0,01
y - целая часть
Теперь используя эту формулу мы переведем ее в обыкновенную:
![Y+ \frac{x1}{1-q}](https://tex.z-dn.net/?f=Y%2B+%5Cfrac%7Bx1%7D%7B1-q%7D+)
Подставим:
![5+ \frac{0,05}{1-0,01}](https://tex.z-dn.net/?f=5%2B+%5Cfrac%7B0%2C05%7D%7B1-0%2C01%7D+)
Решим:
...=
![5 \frac{5}{99}](https://tex.z-dn.net/?f=5+%5Cfrac%7B5%7D%7B99%7D+)
1)Решаем через сочетания элементов:
![P=\frac{C^3_{10}*C^3_5}{C^6_{15}}=\frac{\frac{10!}{7!3!}*\frac{5!}{2!3!}}{\frac{15!}{9!6!}}=\frac{\frac{2*3*4*5*6*7*8*9*10}{2*3*4*5*6*7*2*3}*\frac{2*3*4*5}{2*2*3}}{\frac{2*3*4*5*6*7*8*9*10*11*12*13*14*15}{2*3*4*5*6*7*8*9*2*3*4*5*6}}=\\\\=\frac{1200}{5005}\approx0,24\approx24\%](https://tex.z-dn.net/?f=P%3D%5Cfrac%7BC%5E3_%7B10%7D%2AC%5E3_5%7D%7BC%5E6_%7B15%7D%7D%3D%5Cfrac%7B%5Cfrac%7B10%21%7D%7B7%213%21%7D%2A%5Cfrac%7B5%21%7D%7B2%213%21%7D%7D%7B%5Cfrac%7B15%21%7D%7B9%216%21%7D%7D%3D%5Cfrac%7B%5Cfrac%7B2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A10%7D%7B2%2A3%2A4%2A5%2A6%2A7%2A2%2A3%7D%2A%5Cfrac%7B2%2A3%2A4%2A5%7D%7B2%2A2%2A3%7D%7D%7B%5Cfrac%7B2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A10%2A11%2A12%2A13%2A14%2A15%7D%7B2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A2%2A3%2A4%2A5%2A6%7D%7D%3D%5C%5C%5C%5C%3D%5Cfrac%7B1200%7D%7B5005%7D%5Capprox0%2C24%5Capprox24%5C%25)
2)Решаем через сочетания элементов:
![P=1-\frac{C^6_{10}}{C^6_{15}}=1-\frac{\frac{10!}{4!6!}}{\frac{15!}{9!6!}}=1-\frac{\frac{2*3*4*5*6*7*8*9*10}{2*3*4*2*3*4*5*6}}{\frac{2*3*4*5*6*7*8*9*10*11*12*13*14*15}{2*3*4*5*6*7*8*9*2*3*4*5*6}}=\\=1-\frac{210}{5005}\approx0,96\approx96\%](https://tex.z-dn.net/?f=P%3D1-%5Cfrac%7BC%5E6_%7B10%7D%7D%7BC%5E6_%7B15%7D%7D%3D1-%5Cfrac%7B%5Cfrac%7B10%21%7D%7B4%216%21%7D%7D%7B%5Cfrac%7B15%21%7D%7B9%216%21%7D%7D%3D1-%5Cfrac%7B%5Cfrac%7B2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A10%7D%7B2%2A3%2A4%2A2%2A3%2A4%2A5%2A6%7D%7D%7B%5Cfrac%7B2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A10%2A11%2A12%2A13%2A14%2A15%7D%7B2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A2%2A3%2A4%2A5%2A6%7D%7D%3D%5C%5C%3D1-%5Cfrac%7B210%7D%7B5005%7D%5Capprox0%2C96%5Capprox96%5C%25)