TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2019W/1.1

I Car licence plates

A certain state's car licence plates have three letters of the alphabet followed by a three-digit number.

(a) How many different licence plates are possible if all three-letter sequences are permitted and any number from 000 to 999 is allowed?

(b) Mary withnessed a hit-and-run accident. She knows that the first letter on the licence plate of the offender’s car was a B, that the second letter was an O or Q, and that the last number was a 5. How many state’s licence plates fit this description?

II Symphony orchestra program

A symphony orchestra has in its repertoire 30 Haydn symphonies, 15 modern works, and 9 Beethoven symphonies. Its program always consists of a Haydn symphony followed by a modern work, and then a Beethoven symphony.

(a) How many different programs can it play?

(b) How many different programs are there if the three pieces can be played in any order?

(c) How many different three-piece programs are there if more than one piece from the same category can be played and they can be played in any order?

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Lösungsvorschlag von Gittenburg

--Gittenburg 22:16, 12. Okt. 2019 (CEST)

Car licence plates

a) $26^{3}\cdot 10^{3}=17,576,000$

b) $1\cdot 2\cdot 26\cdot 10^{2}\cdot 1=5,200$

Symphony orchestra program

a) $30\cdot 15\cdot 9=4,050$

b) $30\cdot 15\cdot 9\cdot 3!=24,300$

c) ${\binom {54}{3}}\cdot 3!={\frac {54!}{3!\cdot 51!}}\cdot 3!=54\cdot 53\cdot 52=148,824$