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# Help with counting with restrictions

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1. How many three-digit whole numbers have no 7's and no 9's as digits?

2. Three schools have a chess tournament. Four players come from each school. Each player plays three games against each player from the other schools, and plays one game against each other player from his or her own school. How many games of chess are played?

3. A teacher wants to arrange 3 copies of Introduction to Geometry and 4 copies of Introduction to Number Theory on a bookshelf. In how many ways can he do that?

Jun 10, 2018

#1
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Question 2: every player play 24 games and 3 games against their teammates

thats 27 games

27*12 but this is counting double so we divide by 2

(27*12)/2=162

Jun 10, 2018
#2
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1) Since you cannot use 7s and 9s, therefore we have: 0, 1, 2, 3, 4, 5, 6, 8. Since repeats are allowed, then we have: 8^3 - 8^2 = 448 - 3-digit numbers that have no 7s or 9s in them.

Here is a list of them all:

{1, 0, 0} | {1, 0, 1} | {1, 0, 2} | {1, 0, 3} | {1, 0, 4} | {1, 0, 5} | {1, 0, 6} | {1, 0, 8} | {1, 1, 0} | {1, 1, 1} | {1, 1, 2} | {1, 1, 3} | {1, 1, 4} | {1, 1, 5} | {1, 1, 6} | {1, 1, 8} | {1, 2, 0} | {1, 2, 1} | {1, 2, 2} | {1, 2, 3} | {1, 2, 4} | {1, 2, 5} | {1, 2, 6} | {1, 2, 8} | {1, 3, 0} | {1, 3, 1} | {1, 3, 2} | {1, 3, 3} | {1, 3, 4} | {1, 3, 5} | {1, 3, 6} | {1, 3, 8} | {1, 4, 0} | {1, 4, 1} | {1, 4, 2} | {1, 4, 3} | {1, 4, 4} | {1, 4, 5} | {1, 4, 6} | {1, 4, 8} | {1, 5, 0} | {1, 5, 1} | {1, 5, 2} | {1, 5, 3} | {1, 5, 4} | {1, 5, 5} | {1, 5, 6} | {1, 5, 8} | {1, 6, 0} | {1, 6, 1} | {1, 6, 2} | {1, 6, 3} | {1, 6, 4} | {1, 6, 5} | {1, 6, 6} | {1, 6, 8} | {1, 8, 0} | {1, 8, 1} | {1, 8, 2} | {1, 8, 3} | {1, 8, 4} | {1, 8, 5} | {1, 8, 6} | {1, 8, 8} | {2, 0, 0} | {2, 0, 1} | {2, 0, 2} | {2, 0, 3} | {2, 0, 4} | {2, 0, 5} | {2, 0, 6} | {2, 0, 8} | {2, 1, 0} | {2, 1, 1} | {2, 1, 2} | {2, 1, 3} | {2, 1, 4} | {2, 1, 5} | {2, 1, 6} | {2, 1, 8} | {2, 2, 0} | {2, 2, 1} | {2, 2, 2} | {2, 2, 3} | {2, 2, 4} | {2, 2, 5} | {2, 2, 6} | {2, 2, 8} | {2, 3, 0} | {2, 3, 1} | {2, 3, 2} | {2, 3, 3} | {2, 3, 4} | {2, 3, 5} | {2, 3, 6} | {2, 3, 8} | {2, 4, 0} | {2, 4, 1} | {2, 4, 2} | {2, 4, 3} | {2, 4, 4} | {2, 4, 5} | {2, 4, 6} | {2, 4, 8} | {2, 5, 0} | {2, 5, 1} | {2, 5, 2} | {2, 5, 3} | {2, 5, 4} | {2, 5, 5} | {2, 5, 6} | {2, 5, 8} | {2, 6, 0} | {2, 6, 1} | {2, 6, 2} | {2, 6, 3} | {2, 6, 4} | {2, 6, 5} | {2, 6, 6} | {2, 6, 8} | {2, 8, 0} | {2, 8, 1} | {2, 8, 2} | {2, 8, 3} | {2, 8, 4} | {2, 8, 5} | {2, 8, 6} | {2, 8, 8} | {3, 0, 0} | {3, 0, 1} | {3, 0, 2} | {3, 0, 3} | {3, 0, 4} | {3, 0, 5} | {3, 0, 6} | {3, 0, 8} | {3, 1, 0} | {3, 1, 1} | {3, 1, 2} | {3, 1, 3} | {3, 1, 4} | {3, 1, 5} | {3, 1, 6} | {3, 1, 8} | {3, 2, 0} | {3, 2, 1} | {3, 2, 2} | {3, 2, 3} | {3, 2, 4} | {3, 2, 5} | {3, 2, 6} | {3, 2, 8} | {3, 3, 0} | {3, 3, 1} | {3, 3, 2} | {3, 3, 3} | {3, 3, 4} | {3, 3, 5} | {3, 3, 6} | {3, 3, 8} | {3, 4, 0} | {3, 4, 1} | {3, 4, 2} | {3, 4, 3} | {3, 4, 4} | {3, 4, 5} | {3, 4, 6} | {3, 4, 8} | {3, 5, 0} | {3, 5, 1} | {3, 5, 2} | {3, 5, 3} | {3, 5, 4} | {3, 5, 5} | {3, 5, 6} | {3, 5, 8} | {3, 6, 0} | {3, 6, 1} | {3, 6, 2} | {3, 6, 3} | {3, 6, 4} | {3, 6, 5} | {3, 6, 6} | {3, 6, 8} | {3, 8, 0} | {3, 8, 1} | {3, 8, 2} | {3, 8, 3} | {3, 8, 4} | {3, 8, 5} | {3, 8, 6} | {3, 8, 8} | {4, 0, 0} | {4, 0, 1} | {4, 0, 2} | {4, 0, 3} | {4, 0, 4} | {4, 0, 5} | {4, 0, 6} | {4, 0, 8} | {4, 1, 0} | {4, 1, 1} | {4, 1, 2} | {4, 1, 3} | {4, 1, 4} | {4, 1, 5} | {4, 1, 6} | {4, 1, 8} | {4, 2, 0} | {4, 2, 1} | {4, 2, 2} | {4, 2, 3} | {4, 2, 4} | {4, 2, 5} | {4, 2, 6} | {4, 2, 8} | {4, 3, 0} | {4, 3, 1} | {4, 3, 2} | {4, 3, 3} | {4, 3, 4} | {4, 3, 5} | {4, 3, 6} | {4, 3, 8} | {4, 4, 0} | {4, 4, 1} | {4, 4, 2} | {4, 4, 3} | {4, 4, 4} | {4, 4, 5} | {4, 4, 6} | {4, 4, 8} | {4, 5, 0} | {4, 5, 1} | {4, 5, 2} | {4, 5, 3} | {4, 5, 4} | {4, 5, 5} | {4, 5, 6} | {4, 5, 8} | {4, 6, 0} | {4, 6, 1} | {4, 6, 2} | {4, 6, 3} | {4, 6, 4} | {4, 6, 5} | {4, 6, 6} | {4, 6, 8} | {4, 8, 0} | {4, 8, 1} | {4, 8, 2} | {4, 8, 3} | {4, 8, 4} | {4, 8, 5} | {4, 8, 6} | {4, 8, 8} | {5, 0, 0} | {5, 0, 1} | {5, 0, 2} | {5, 0, 3} | {5, 0, 4} | {5, 0, 5} | {5, 0, 6} | {5, 0, 8} | {5, 1, 0} | {5, 1, 1} | {5, 1, 2} | {5, 1, 3} | {5, 1, 4} | {5, 1, 5} | {5, 1, 6} | {5, 1, 8} | {5, 2, 0} | {5, 2, 1} | {5, 2, 2} | {5, 2, 3} | {5, 2, 4} | {5, 2, 5} | {5, 2, 6} | {5, 2, 8} | {5, 3, 0} | {5, 3, 1} | {5, 3, 2} | {5, 3, 3} | {5, 3, 4} | {5, 3, 5} | {5, 3, 6} | {5, 3, 8} | {5, 4, 0} | {5, 4, 1} | {5, 4, 2} | {5, 4, 3} | {5, 4, 4} | {5, 4, 5} | {5, 4, 6} | {5, 4, 8} | {5, 5, 0} | {5, 5, 1} | {5, 5, 2} | {5, 5, 3} | {5, 5, 4} | {5, 5, 5} | {5, 5, 6} | {5, 5, 8} | {5, 6, 0} | {5, 6, 1} | {5, 6, 2} | {5, 6, 3} | {5, 6, 4} | {5, 6, 5} | {5, 6, 6} | {5, 6, 8} | {5, 8, 0} | {5, 8, 1} | {5, 8, 2} | {5, 8, 3} | {5, 8, 4} | {5, 8, 5} | {5, 8, 6} | {5, 8, 8} | {6, 0, 0} | {6, 0, 1} | {6, 0, 2} | {6, 0, 3} | {6, 0, 4} | {6, 0, 5} | {6, 0, 6} | {6, 0, 8} | {6, 1, 0} | {6, 1, 1} | {6, 1, 2} | {6, 1, 3} | {6, 1, 4} | {6, 1, 5} | {6, 1, 6} | {6, 1, 8} | {6, 2, 0} | {6, 2, 1} | {6, 2, 2} | {6, 2, 3} | {6, 2, 4} | {6, 2, 5} | {6, 2, 6} | {6, 2, 8} | {6, 3, 0} | {6, 3, 1} | {6, 3, 2} | {6, 3, 3} | {6, 3, 4} | {6, 3, 5} | {6, 3, 6} | {6, 3, 8} | {6, 4, 0} | {6, 4, 1} | {6, 4, 2} | {6, 4, 3} | {6, 4, 4} | {6, 4, 5} | {6, 4, 6} | {6, 4, 8} | {6, 5, 0} | {6, 5, 1} | {6, 5, 2} | {6, 5, 3} | {6, 5, 4} | {6, 5, 5} | {6, 5, 6} | {6, 5, 8} | {6, 6, 0} | {6, 6, 1} | {6, 6, 2} | {6, 6, 3} | {6, 6, 4} | {6, 6, 5} | {6, 6, 6} | {6, 6, 8} | {6, 8, 0} | {6, 8, 1} | {6, 8, 2} | {6, 8, 3} | {6, 8, 4} | {6, 8, 5} | {6, 8, 6} | {6, 8, 8} | {8, 0, 0} | {8, 0, 1} | {8, 0, 2} | {8, 0, 3} | {8, 0, 4} | {8, 0, 5} | {8, 0, 6} | {8, 0, 8} | {8, 1, 0} | {8, 1, 1} | {8, 1, 2} | {8, 1, 3} | {8, 1, 4} | {8, 1, 5} | {8, 1, 6} | {8, 1, 8} | {8, 2, 0} | {8, 2, 1} | {8, 2, 2} | {8, 2, 3} | {8, 2, 4} | {8, 2, 5} | {8, 2, 6} | {8, 2, 8} | {8, 3, 0} | {8, 3, 1} | {8, 3, 2} | {8, 3, 3} | {8, 3, 4} | {8, 3, 5} | {8, 3, 6} | {8, 3, 8} | {8, 4, 0} | {8, 4, 1} | {8, 4, 2} | {8, 4, 3} | {8, 4, 4} | {8, 4, 5} | {8, 4, 6} | {8, 4, 8} | {8, 5, 0} | {8, 5, 1} | {8, 5, 2} | {8, 5, 3} | {8, 5, 4} | {8, 5, 5} | {8, 5, 6} | {8, 5, 8} | {8, 6, 0} | {8, 6, 1} | {8, 6, 2} | {8, 6, 3} | {8, 6, 4} | {8, 6, 5} | {8, 6, 6} | {8, 6, 8} | {8, 8, 0} | {8, 8, 1} | {8, 8, 2} | {8, 8, 3} | {8, 8, 4} | {8, 8, 5} | {8, 8, 6} | {8, 8, 8} (total: 448)

Jun 11, 2018