(related to Problem: The Barrels Of Balsam)
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, this: $$\frac{\binom{2n}n}{n + 1}$$ where $2n$ equals the number of barrels. The symbol $\binom{2n}n$, of course, implies that we have to find how many combinations, or selections, we can make of $2n$ things, taken $n$ at a time.
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