Dean Chalmers and Julian Gilbey Solutions for Chapter: Permutations and Combinations, Exercise 12: END-OF-CHAPTER REVIEW EXERCISE 5

Twenty teams have entered a tournament. In order to reduce the number of teams to eight, they are put into groups of five and the teams in each group play each other twice. The top two teams in each group progress to the next round. From this point on, teams are paired up, playing each other once with the losing team being eliminated. How many games are played during the whole tournament?

Five cards, each marked with a different single-digit number from $3$ to $7,$ are randomly placed in a row. Find the probability that the first card in the row is odd and that the three cards in the middle of the row have a sum of $15.$

How many even four-digit numbers can be made from the digits $0,2,3,4,5$ and $7,$ each used at most once, when the first digit cannot be zero?

Find how many numbers there are between $100$ and $999$ in which all three digits are different.

Of the numbers between $100$ and $999$ in which all three digits are different. Find how many of the numbers are odd numbers greater than$700.$

A bunch of flowers consists of a mixture of roses, tulips and daffodils. Tom orders a bunch of $7$ flowers from a shop to give to a friend. There must be at least $2$ of each type of flower. The shop has $6$ roses, $5$ tulips and $4$ daffodils, all different from each other. Find the number of different bunches of flowers that are possible.

Three identical cans of cola, $2$ identical cans of green tea and $2$ identical cans of orange juice are arranged in a row. Calculate the number of arrangements if

the first and last cans in the row are the same type of drink,

Three identical cans of cola, $2$ identical cans of green tea and $2$ identical cans of orange juice are arranged in a row. Calculate the number of arrangements if

the $3$ cans of cola are all next to each other and the $2$ cans of green tea are not next to each other.