Exercises: The fundamental principle of counting

A café in Kilkenny offers $3$ types of sandwich and $4$ types of drink. How many different meals consisting of one sandwich and one drink are possible?

A school jumper comes in $5$ sizes and $2$ colours. How many different jumpers are available?

A coin is tossed and then a standard six-sided die is rolled. How many different outcomes are possible?

A locker code uses $1$ letter from A–F followed by $2$ digits from $0$–$9$ (digits may repeat).

1. a) How many codes are possible?

1. b) How many codes are possible if the digits must be different?

Seán has $4$ T-shirts, $3$ pairs of shorts and $2$ pairs of runners.

1. a) How many different outfits can he make?

1. b) How many if he refuses to wear his red T-shirt with his green shorts?

Five runners, Aoife, Brian, Ciara, Dara and Éabha, finish a race with no ties.

1. a) How many ways can the first three places be filled?

1. b) In how many of these is Ciara first?

How many three-digit numbers can be made from the digits $1, 2, 3, 4, 5$ if

1. a) digits can be repeated?

1. b) digits cannot be repeated?

1. c) digits cannot be repeated and the number is even?

A car registration plate in a game has $2$ letters from A–Z followed by $3$ digits from $0$–$9$.

1. a) How many plates are possible if repetition is allowed?

1. b) How many have all five characters different?

1. c) Explain why your answer to b) is smaller than a).

A committee of a chairperson, a secretary and a treasurer is chosen from $8$ students, and no student may hold two roles.

1. a) How many different committees are possible?

1. b) If Niamh must be chosen for one of the three roles, how many committees are possible?

1. c) Show that the number of committees without Niamh plus the number with her equals your answer to a).