Track / Stage 4 / 300 of 340 #560 of 1964

Problem 560

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Find the answer

23. On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are of 101,102 , $103,104,105$ and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
A $75 \%$
B $80 \%$
C $90 \%$
D $95 \%$
E $100 \%$

Your answer

Multiple choice: answer with the letter of the option you want.

Official solution

23. B The total mass is $621 \mathrm{~g}$ so any three masses with total mass over $310.5 \mathrm{~g}$ could be in the heavier pan. There are eight of these triples that include the $106 \mathrm{~g}$ mass: $(106,105,104)$ , $(106,105,103),(106,105,102),(106,105,101),(106,104,103),(106,104,102),(106$ , $104,101)$ , and $(106,103,102)$ .
Without 106 , there are 2 ways to make a set over $310.5 \mathrm{~g}$ : $(105,104,103)$ and $(105,104$ , 102 ).
Hence the probabililty that the $106 \mathrm{~g}$ mass is included in the heavier pan is $\frac{8}{8+2}=\frac{8}{10}$ or $80 \%$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.