This is quite easy to solve for any number of
barrels—if you know how.
This is the way to do it.
There are five barrels in each row Multiply the numbers 1, 2, 3, 4, 5
together; and also multiply 6, 7, 8, 9, 10 together.
Divide one result by the other, and we get the number of different
combinations or selections of ten things taken five at a time.
This is here 252.
Now, if we divide this by 6 (1 more than the number in the row) we get
42, which is the correct answer to the puzzle, for there are 42
different ways of arranging the barrels.
Try this method of solution in the case of six barrels, three in each
row, and you will find the answer is 5 ways.
If you check this by trial, you will discover the five arrangements
with 123, 124, 125, 134, 135 respectively in the top row, and you will
find no others.
The general solution to the problem is, in fact,
equals the number of
The symbol C, of course, implies that we have to
find how many combinations, or selections, we can make of 2n
things, taken n at a time.