(related to Problem: The Excursion Ticket Puzzle)
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 labor 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 $$\frac{2n^3+15n^2+33n}{18} + 1$$
ways. If sixpences are also used there are $$\frac{n^4+22n^3+159n^2+414n+216}{216}$$
ways, when the sum is a multiple of sixpence, and the constant, 216, changes to 324 when the money is not such a multiple. And so the formulas increase in complexity in an accelerating ratio as we go on to the other coins.
I will, however, add an interesting little table of the possible ways of changing our current coins which I believe has never been given in a book before. Change may be given for a * Farthing in $0$ way. * Halfpenny in $1$ way. * Penny in $3$ ways. * Threepenny-piece in $16$ ways. * Sixpence in $66$ ways. * Shilling in $402$ ways. * Florin in $3,818$ ways. * Half-crown in $8,709$ ways. * Double florin in $60,239$ ways. * Crown in $166,651$ ways. * Half-sovereign in $6,261,622$ ways. * Sovereign in $500,291,833$ ways.
It is a little surprising to find that a sovereign may be changed in over five hundred million different ways. But I have no doubt as to the correctness of my figures.
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