Question

Решите пожалуйста
даю 55 баллов

Answer 1

Объяснение:

$a)\ \frac{P_{20}}{A_{20}^{15}}-\frac{A_5^{20}}{C_5^{20}} =\frac{20!}{\frac{20!!}{(20-15)!} } -\frac{\frac{20!}{(20-5)!} }{\frac{20!}{(20-5)!*5!} }=5!-5!=0.\\b)\ \frac{A_{12}^4-A_{11}^4}{C^3_{11}} =\frac{\frac{12!}{(12-4)!}-\frac{11}{(11-4)!} }{\frac{11!}{(11-3)!*3!} } =\frac{\frac{8!*9*10*11*12}{8!} -\frac{7!*8*9*10*11}{7!} }{\frac{8!*9*10*11}{8! *1*2*3}}=\frac{9*10*11*12-8*9*10*11}{\frac{9*10*11}{6} }=\\=\frac{6*(9*10*11)*(12-8)}{9*10*11} =6*4=24.$

$c)\ \frac{A_{18}^{10} }{C_{18}^{10} } -\frac{P_{18}}{A_{18}^8} =\frac{\frac{18!}{(18-10)!} }{\frac{18!}{(18-10)!*10!} } -\frac{18!}{\frac{18!}{(18-8)!} } =10!-10!=0.\\d)\ \frac{A_{15}^5-A_{14}^5}{C_{14}^4}=\frac{\frac{15!}{(15-5)!} -\frac{14!}{(14-5)!} }{\frac{14!}{(14-4)!*4!} }=\frac{\frac{10!*11*12*13*14*15}{10!} -\frac{9!*10*11*12*13*14}{9!} }{\frac{10!*11*12*13*14}{10!*1*2*3*4} } =\frac{11*12*13*14*15-10*11*12*13*14}{\frac{11*12*13*14}{24} } =$$=\frac{11*12*13*14*(15-10)*24}{11*12*13*14}=5*24=120.$

Answer 2

Ответ:

$1)\ \ \dfrac{P_{20}}{A_{20}^{15}}-\dfrac{A_{20}^{5}}{C_{20}^{5}}=\dfrac{20!}{\dfrac{20!}{5!}}-\dfrac{\dfrac{20!}{15!}}{\dfrac{20!}{15!\, 5!}}=5!-5!=0\\\\\\2)\ \dfrac{A_{12}^4-A_{11}^4}{C_{11}^3}=\dfrac{(12\cdot 11\cdot 10\cdot 9)-(11\cdot 10\cdot 9\cdot 8)}{\dfrac{11\cdot 10\cdot 9}{3!}}=\dfrac{11\cdot 10\cdot 9\cdot (12-8)}{\dfrac{11\cdot 10\cdot 9}{2\cdot 3}}=\\\\\\=4\cdot 2\cdot 3=4!=24$

$3)\ \ \dfrac{A_{18}^{10}}{C_{18}^{10}}-\dfrac{P_{18}}{A_{18}^8}=\dfrac{C_{18}^{10}\cdot P_{10}}{C_{18}^{10}}-\dfrac{18!}{\dfrac{18!}{10!}}=10!-10!=0\\\\\\4)\ \ \dfrac{A_{15}^5-A_{14}^5}{C_{14}^4}=\dfrac{15\cdot 14\cdot 13\cdot 12\cdot 11-14\cdot 13\cdot 12\cdot 11\cdot 10}{\dfrac{14\cdot 13\cdot 12\cdot 11}{4!}}=\\\\\\=\dfrac{14\cdot 13\cdot 12\cdot 11\cdot (15-10)\cdot 4!}{14\cdot 13\cdot 12\cdot 11}=5\cdot 4!=5!=120$