A PACK OF CARDS.

New Zealand Herald, Volume XLVIII, Issue 14714, 23 June 1911, Page 4

A PACK OF CARDS.

THE POSSIBLE COMBINATIONS.

A pack of cards! This is a- common enough object- surely. It would lax the resources of even the ablest- calculator to estimate the amount of time which has been spent by men and women in eagerly watching the varying chances of gain or loss, or maybe merely of amusement, due to the multitudinous combinations of these 52 pieces of coloured pasteboard. But however keen the players may be, especially when a lurgo sum of money is at stake, in their endeavours to judge correctly the chances of any particular lino of play, it remains a- fact that the total possible number of combinations of this comparatively small number of cards is so vast that it becomes impossible to frame a theory covering all cases, even when the attempt is aided by classifying the cases. As one instance of this, the fairly well-known fact may bo mentioned that in a game like whist or bridge, where each player receives a hand of 13 cards, selected from the 52, tho total possible number of different hands which may thus be selected is greater than six hundred thousand million. The exact number is 655,013,559,600. This number, however, large though it may appear, becomes not merely small, but absolutely invisible, when compared with the total number of arrangements in which the 52 cards may be placed after the shuttle. To quote the whole of this number would take too much space, but it may bo mentioned that it begins with 80, followed by 66 other figures. It is probably a quite hopeless task to attempt to enable anyone to grasp the conception of the real meaning of such a number ; but some idea (at least of its inconceivability) may be presented by means of the following calculations. Let us suppose that two thousand millions of human beings (each supplied with a pack of cards) were to attempt actually to produce every possible arrangement of the 52 cards. It is further to be supposed that they work ceaselessly, without rest day or night, from year's end to year's end, at the rate of one now arrangement- per second for each person during a period of 100,000 years. It should be noted that the entire population of the earth to-day is estimated to be in the neighbourhood of 1600 millions. The hypothesis from which we start is therefore thai; a population one quarter more than that which now exists has spentits whole time during an interval more than 50 times the duration of the Christian era in shuffling cards at the rate of one shuffle per second, or more than 31 million shuffles in each year per head. In view of such (inures, the reader may well ask how many times the total number of arrangements will have been produced by this vast amount of sustained (though illdirected) human effort. Tho answer is not once. Calculation proves in fact that the number of card arrangements produced tinder the conditions assumed will only bo a. minute fraction of the total possible number—a fraction so minute, in fact, that it becomes necessary to devise another scheme of concrete representation in order to give an idea of its minuteness.

Drops of Water. Let it be assumed that the whole vast number of arrangements produced by the human race as above is symbolised by one drop of water.* Then how much water would be required to symbolise the total number possible? If this question is put to the reader, he might well say : Surely a glass of water would be enough But no. "A bathful of water then?" No. "A large reservoir?" No, my friend, you must enlarge your conceptions, or you will never reach the truth. "Tho Atlantic Ocean, then? The number of drops in that will surely be sufficient?" But the number of drops of water in the Atlantic Ocean is not sufficient, nor will it become so, even when we add to the Atlantic Ocean the Pacific, and all the other oceans, seas, lakes, and in fact all the rest of the water on our globe. Nor would tho whole earth, made from centre to surface entirely of water, bo sufficient; nor would the whole sun similarly constituted suffice. Incrediblo though it may seem, to obtain a volume of water containing a sufficient number of dr' ps, it is necessary to imagine a globe of water with a diameter equal to 7025 .nillions of miles.

The Summing-up. If the centre of such a globe is taken at the centre of tho sun, then Neptune, the remotest planet in the solar system, would be immersed therein to a depth of no less than 700 millions of miles; in other words, such a ball of water would have a diameter about 25 per cent, greater than that of the whole solar system as at present known. In addition to which, remember, the solar system is practically included in a flat disc, with extension in one plane only, whereas tho ball to which we have been so unexpectedly Jed would extend not only in that plane, but also, and to an equal extent, upwards and downwards from it; the relation, in fact, would be that of a cricket ball to a biscuit. After the evidence comes the summing-up. Thus, firstly, we take one drop of water to represent the whole result of the sustained energy of the human race (or, indeed, more than tho present number thereof) directed at break-nock speed to one only object, throughout an interval of time extending not into the dawn of history but beyond it into a period when man had probably not established his mastery over the animal creation. And, secondly, we find that though this one drop of water represents a result so enormous that it can hardly bo grasped, yet the total result to bo obtained is still so far off that to represent it on the same scale, requires what may almost be described as a watery universe. Calculations. Tho method by which the total number of arrangements of tho 52. cards is calculated can be readily explained. For any one of the 52 may bo chosen as the top card, and when some card has been chosen, then any one of tho 51 remaining cards can be chosen as second card, so that the two first cards may bo chosen in order in 52 times 51, or 2652 ways. Thero then remain 50 cards any one of which may be selected as third card, from which it results that the three first cards can be chosen in order in 50 times, 2652, or 132,600 ways. Proceeding thus, it is evident that, the total number of arrangements of the 52 cards can be calculated by multiplying together all the numbers 52, 51, 50, etc., down to 1. Such a calculation presents no difficulty but its length and tediousness : the result is a number of 68 figures, the last 12 of which aro zeros. The rest of tho calculation, to show that, 011 (he scale adopted, <t globe of water with a diameter larger than that of the orbit of Neptune would bo needed to represent tho total number of .arrangements, is given below. It should be mentioned that tho drop" used as the unit in this illustration is flie chemist's drop, or " minim," of which 480 go to tho ounce, or 76,000 to tho gallon. Drops vary in size according to the conditions under which they aro produced ; in an actual, experiment in filling a one-ounce measure from a drop-ping-bottle, 399 drops were needed ; thus these drops were 20 per cent, larger than minims, and with such a drop as the basis the volume of the immense globe of water we are considering would be correspondingly greater. . The calculations arc, of course, done by means of logarithms, correct to seven decimals. _ When it is stated that some quantity is "more than" the number given, the meaning is that such number is correct ;is far as it goes, but that further figures would have to bo added at the end in order to set it out- in full.—(J. A. Rossetti.