Answer to Question #323672 in Discrete Mathematics for gift

Question

1, In how many ways each 5 boys and 3 girls be seated around a table if:

i) There is no restriction?

ii, Boy Bi and girl Gi are not adjacent?

ji, No girls are adjacent?

2. Each user on a computer system has a password, which is six to eight characters

long. where each character is an upper case letter or a digit Fach password must

contain at least one digit. How many possible passwords are there?

3.how many ways are there to distribute 10 identical bones to 4 dogs if each dog must

stat least | bone and Fifi may not receive more than 3 bones’?

Answer 1

$i:\\8!=40320\\ii:\\Find\,\,the\,\,variants\,\,when\,\,Bi\,\,and\,\,Gi\,\,are\,\,adjacent:\\8 variants\,\,to\,\,choose\,\,a\,\,pair\,\,of\,\,seats\,\,for\,\,them\,\,\left( we\,\,choose\,\,the\,\,chair\,\,which\,\,is\,\,to\,\,the\,\,left\,\,if\,\,walking\,\,clockwise \right) ,\\2 variants\,\,to\,\,seat\,\,them,\\6!=720\,\,variants\,\,to\,\,seat\,\,other\,\,kids.That\,\,is\,\,\\8\cdot 2\cdot 720=11520\\variants.\\Then\,\,the\,\,answer\,\,is\\40320-11520=28800\\variants.\\ji:\\The\,\,possible\,\,chairs\,\,for\,\,girls\,\,are\\1,3,5\\1,3,6\\1,3,7\\1,4,6\\1,4,7\\1,5,7\\2,4,6\\2,4,7\\2,4,8\\2,5,7\\2,5,8\\2,6,8\\3,5,7\\3,5,8\\3,6,8\\4,6,8\\Thus\,\,there\,\,are\,\,16ways\,\,to\,\,seat\,\,the\,\,girls.\\3!=6\,\,variants\,\,of\,\,seating\,\,girls.\\5!=120\,\,variants\,\,of\,\,seating\,\,boys.\\Total\,\,is\\16\cdot 6\cdot 120=11520$

$26 letters\\10 digits\\2:\\Passwords\,\,of\,\,6 digits: \\P_{36}^{6}-P_{26}^{6}=\frac{36!}{30!}-\frac{26!}{20!}\\Passwords\,\,of\,\,7 digits: \\P_{36}^{7}-P_{26}^{7}=\frac{36!}{29!}-\frac{26!}{19!}\\Passwords\,\,of\,\,8 digits: \\P_{36}^{8}-P_{26}^{8}=\frac{36!}{28!}-\frac{26!}{18!}\\Total\\\frac{36!}{30!}-\frac{26!}{20!}+\frac{36!}{29!}-\frac{26!}{19!}+\frac{36!}{28!}-\frac{26!}{18!}=1.1971\times 10^{12}$