Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 1178. (September 2013)

C. 1178. Bill wants to buy a glass of soda from a vending machine for 60 forints (HUF, Hungarian currency). He has five 10-forint coins and four 20-forint coins in his pocket. He pulls out coins at random. What is the probability that by pulling out four coins in a row he will get exactly 60 forints from his pocket?

(5 pont)

Deadline expired on October 10, 2013.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Akkor lesz négy húzással 60 Ft a kezében, ha kétszer 10, kétszer pedig 20 forintot húz. Mivel 10 Ft-osból 5 darab, 20 Ft-osból pedig 4 darab van, ezért ezt \(\displaystyle \binom 52\cdot\binom42=10\cdot6=60\)-féleképpen teheti meg. Az összes esetet úgy kapjuk meg, ha összeadjuk azt, mikor 0 darab 10 és 4 darab 20 forintost húz, 1 darab 10 és 3 darab 20 forintost húz stb. Vagyis az összes eset száma:

\(\displaystyle \binom50\binom44+\binom51\binom43+\binom52\binom42+\binom53\binom41+\binom54\binom40= 1\cdot1+5\cdot4+10\cdot6+10\cdot4+5\cdot1=126.\)

A valószínűség: \(\displaystyle \frac{60}{126}=\frac{10}{21}\approx0,4762\).


Statistics:

378 students sent a solution.
5 points:203 students.
4 points:10 students.
3 points:21 students.
2 points:125 students.
1 point:10 students.
0 point:7 students.
Unfair, not evaluated:2 solutions.

Problems in Mathematics of KöMaL, September 2013