6 balls: 2 weigh 1 unit each, 2 weigh 2 units each and 2 weigh 3 units each.

The 6 balls are placed at random on a balance scale: 3 on each side.

What is the probability that the the scale will balance?

1 in 15 is what I get.

a) 720 permutations total (6!)
b) it will only balance with a 1 2 3 on each side
c) call the balls 1a 1b 2a 2b 3a 3b then there are 6 permutations of any set of 3 balls on one side (3!)
d) there are 8 combinations of a balls or b balls taken in a group of 1 2 and 3.
e) 6*8=48
f) 48 in 720 = 1 in 15

now I hope I didn't make a mistake and my assumptions were correct.

Sorry Bob, but too low.

Hint: probability is more than 1/3

2/5

(6 C 3) = 20

As bob said, there are 8 combinations with 1,2,3
on each side of the scale.

8/20 = 2/5

aha, I didn't account for the fact that you are really just picking 3 of the 6 to be on one side.

That is surprisingly high.

2/5 correct.

Easier still:
since 1,2,3 is a must, then:
1: 1 or 2 or 3: probability = 1
2: (assume 1 was 1st) 2 or 3: prob. = 4/5
3: (assume 2 was 2nd) 3 : prob. = 1/2

1 * 4/5 * 1/2 = 2/5