Irish Leaving Certificate Mathematics

Counting problems ask how many outcomes an experiment has, or how many ways something can be arranged or chosen, without listing every possibility. The same ideas underlie the probability of equally likely outcomes, where the probability of an event is the number of outcomes in the event divided by the total number of outcomes. ## Listing outcomes systematically The outcomes of a simple experiment can be listed in an organised way so that none is missed or repeated. For two coins the outcomes are HH, HT, TH and TT. For two fair dice the outcomes form a $6 \times 6$ table of 36 ordered pairs, where $(2, 5)$ and $(5, 2)$ are different outcomes. Listing in order (all outcomes beginning with the first possibility, then the second, and so on) or drawing a table or tree diagram are the usual ways of keeping a list complete. ## The fundamental principle of counting If one task can be done in $m$ ways and, whichever way it is done, a second task can then be done in $n$ ways, the two tasks together can be done in $m \times n$ ways. The rule extends to any number of tasks done one after another. A restaurant offers 4 starters, 6 main courses and 3 desserts. A full meal with one of each can be chosen in $4 \times 6 \times 3 = 72$ ways. The principle applies just as well when the choices at later stages depend on earlier ones, provided the number of options at each stage is fixed. A PIN of 4 digits, where the first digit cannot be 0 and no digit may repeat, can be formed in $9 \times 9…