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them according to the rank that he has named them, he pays that which each of the players has wagered in the game, & gives the hand to the one who follows him at the right. But if it happens to him in the sequence of thirteen cards, to draw the card which he names, for example, to draw an ace at the time which he names one, or a two at the time which he names two, or a three at the time which he names three, &c. he takes all that which is in the game, & restarts as before, naming one, next two, &c. It is able to happen that Pierre having won many times, & restarting with one, has not enough cards in his hand in order to go up to thirteen, now he must, when the deck falls short to him, to shuffle the cards, to give to cut, & next to draw from the entire deck the number of cards which is necessary to him in order to continue the game, by commencing with the one where he is stopped in the preceding hand. For example, if drawing the last card from them he has named seven, he must in drawing the first card from the entire deck, after one has cut, to name eight, & next nine, &c. up to thirteen, unless he rather not win, in which case he would restart, naming first one, next two, & the rest as it happens in the explanation. Whence it seems that Pierre is able to make many hands in sequence, & likewise he is able to continue the game indefinitely.
    This extract (for which we have relied on the translation by Richard J. Pulskamp) is from the book Essai d’analyse sur les jeux de hazard , 2nd edn (1713), by Pierre Renard de Montmort. It is followed by an analysis which, in true mathematical fashion, starts with easier cases, moving to a full solution (with significant contributions from Nicolas Bernoulli). Subsequently other luminaries considered variations of the problem, including De Moivre, Euler, Lambert and Laplace, and we will consider whatis probably its most common modern form with its most common name of rencontre (a French word which can be translated as ‘meet by chance’), in which the thirteen card limit is replaced by the whole fifty-two cards of the pack. Since it is the form in which Euler considered the problem we will let him explain it with this extract from the article, Calcul de la probabilité dans le jeu de rencontre, which appears in Memoires de l’academie des sciences de Berlin 7, 1753 (again we have relied on the translation of Richard J. Pulskamp):
    The game of rencontre is a game of chance where two persons, each having an entire deck of cards, draw from it at the same time one card after the other, until it happens that they encounter the same card; and then one of the two persons wins. Now, when such an encounter does not happen at all, then it is the other of the two persons who wins. This posed, one asks the probability that each of these two persons will win.
    Whatever the order of the cards in one pack, the other pack will consist of some permutation of those cards and we can look at the game from one of the players’ points of view by considering the chance that no encounter occurs, which brings us to a useful definition.
    Derangements
    A permutation which leaves no element fixed has become known as a derangement ; put another way, a derangement of n objects is a permutation of them without fixed points. The number of derangements of n objects is usually written as ! n , spoken as subfactorial n .
    Of course, not all permutations are derangements: for example, { 5,1,2,3,4} is a permutation of { 1,2,3,4,5} in which no number occupies its original place and so is a derangement, { 5,2,1,3,4} is not, as the 2 remains fixed.
    If we take Montmort’s approach and look at the three simplest cases, we can easily see that the single permutation of { 1} has no derangements, those of { 1,2} have the single derangement { 2,1 } and those of { 1,2,3 } have the two derangements { 2,3,1 } and { 3,1,2 } ; this means that

    In general, we need to ask the question: Of the ! n different permutations of n distinct