8.01 Counting techniques
If there are 5 swimmers in a race, in how many different orders can they finish (assuming there are no ties)?
How many ways can 9 different items be arranged in a line?
How many different ways can the letters of the word 'SHELF' be arranged?
Companies prefer their product to be placed on the middle shelf in supermarkets. If there are 5 brands, but only space for 2 of them on the middle shelf, how many arrangements are possible on the middle shelf?
In how many ways can 8 oarsmen be positioned in a boat if half of them can only row on the stroke side and the other half can only row on the bow side?
A hotel is supervised by a team of 6 security guards at any given time during the day. If there are 12 security guards available, how many different ways can a team of guards be chosen?
In a Lotto draw 5 different numbers are chosen from the numbers 1 to 35. How many possible selections are there? Note that order does not matter.
A boss wants to select one group of 4 people from his 28 staff. How many different groups are possible?
A restaurant offers seven different spaghetti sauces. Tricia chooses two sauces for her spaghetti. How many different possible choices could she make?
A variety pack of chocolate consists of seven bars, each with a different flavour. If three bars of chocolate are chosen at random, how many different selections are possible?
Out of the 16-man cricket touring squad, only 12 are selected for the team.
How many different teams can be made?
Two opening batsmen are to be selected from among the 12 chosen players. How many combinations of opening batsmen are possible?
In court, a jury panel of 8 members is to be made up from a group of 20 candidates. In how many different ways can a panel be formed?
A university has 5 flagpoles and 8 different flags. In how many ways can the flags be arranged on the 5 flagpoles? (Only 1 flag per flagpole and order of flagpoles does matter.)
A board of directors wants to elect a president, secretary and treasurer from its 10 members. In how many ways can the election turn out if each member has an equal chance of being elected to a position and each member can only fill one position?
If there are 26 entrants in a particular poker tournament and only the top 3 get paid, how many different orderings of the paid places are possible?
In a bicycle race, a quinella is a bet on the first 2 bicycles that finish the race, but the order in which these 2 bicycles place does not matter. How many different quinella bets are possible for a bicycle race where 14 bicycles are competing?
2 girls and 3 boys are queuing up to purchase their meals at a canteen.
In how many different ways can the 2 girls and 3 boys line up at the canteen?
In how many ways can they line up if the 2 girls are placed ahead of the 3 boys?
To determine the successful applicant for a job, the interviewer assigns a problem at random to each applicant. If there are 4 applicants, how many different ways can the problems be assigned to them?
A newspaper editor is deciding which of 6 articles to print on the front page.
If she can only choose 2 of them for the front page, how many different selections are possible?
If their order on the front page matters, how many different arrangements are possible for the front page?
She finds an error in one of the 6 articles and cannot print it. How many different arrangements for the front page are now possible, given that the order of the articles on the front page still matters?
In a football squad, there are 5 midfielders. Assuming midfielders usually move in a line and their position in the line is important:
How many arrangements are possible if all the midfielders are used in a game?
How many arrangements are possible if 2 of the midfielders are used in a game?
If their order in the line is not important, how many selections of 2 midfielders are possible?
Evaluate the following expressions:
5!
3!
0!
\dfrac{7!}{5!}
\dfrac{10!}{9!}
\dfrac{6!}{8!}
\left(7 - 5\right)!
\left(11 - 8\right)!
Evaluate the following expressions:
4! 6!
3! 7!
\dfrac{7!}{4!}
\dfrac{6!}{0!}
\dfrac{0!}{9!}
4! \times 5
5\times 3! \times 4
(2 \times 3)!
Evaluate the following expressions:
\dfrac{11!}{5! 6!}
\dfrac{19!}{15! 4!}
\dfrac{6!}{4! 4!}
\dfrac{6!2!}{8!4!}
Express 6! \times 7 as a single factorial in the form x!.
Rewrite each of the following as a single factorial expression:
Rewrite the following as a single factorial expression: \left(n - 1\right)! n \left(n + 1\right) for some positive whole number n.
Express the following without any factorial notation: \dfrac{\left(n + 2\right)!}{\left(n - 2\right)!} for some positive whole number n.