Nineteen shillings and ninepence may be paid in 458,908,622 different
ways.
I do not propose to give my method of solution. Any such explanation
would occupy an amount of space out of proportion to its interest or
value. If I could give within reasonable limits a general solution for
all money payments, I would strain a point to find room; but such a
solution would be extremely complex and cumbersome, and I do not
consider it worth the labour of working out.
Just to give an idea of what such a solution would involve, I will
merely say that I find that, dealing only with those sums of money that
are multiples of threepence, if we only use bronze coins any sum can be
paid in (n + 1)² ways where n always represents the number of
pence. If threepenny-pieces are admitted, there are
2n³ + 15n² + 33n
--------------------- + 1 ways.
18