MEMORIAL TO CLASSICAL STATISTICS 115 



basket II, and so on until every one of the balls is reposing in one or another 

 basket. Now along comes the umpire with pencil and pen, and he writes 

 down on one sheet of his pad of paper the numbers of all the balls which are 

 in basket I, and on a second sheet the numbers of all which are in II, and so 

 on until he has got an inventory of the contents of all of the baskets. The 

 inventory does not state the order in which the balls in any basket were dropped 

 into that basket. That order is blotted out and forgotten. The inventory 

 states which balls are in which baskets, and lets it go at that. 



This does not seem a very entertaining game, but entertainment is not 

 what it is for. The present question is: how many different inventories can 

 there be, consistent with that sequence of figures A^i , iV2 , iVa , • ■ • Nm which 

 the caller prescribed at the start? 



The answer is obtained in what must seem, to anyone meeting for the 

 first time such a question, a strangely devious way. 



First we evaluate the whole number of different orders in which the balls 

 can be drawn from the sack. This is A^-factorial or AH; for the first ball to 

 emerge may be any one of the N, and the next may then be any one of the 

 (A'' — 1) remaining, and the next may then be any one of the {N — 2) remain- 

 ing, and so on to the end. 



If each order corresponded to a different inventory, N\ would be our 

 answer. Clearly this is so, if and only if there are as many baskets as balls 

 and one ball in ever^- basket. In all other cases Nl is larger, and often 

 colossally larger, than the number which we seek. It is necessary now to see 

 that this great multitude of N ! different orders falls into groups composed of 

 -Y orders apiece, all of those in a single group corresponding to a single 

 inventory — necessary to see this, and to calculate X; whereupon we shall 

 find that X, the "number of orders per inventory", is the same for all of the 

 inventories — so that the number which we seek is A^! divided by this common 

 value of X. 



It seems to be helpful to think of some one inventory, and of some one 

 order which leads to that inventory. By a certain amount of mental effort, 

 which varies from person to person, it can be seen that this particular order 

 is but one among Nil N2I N3I • • ■ NmI different orders all leading to the 

 very same inventory. For think of the A^i numbered balls which lie in the 

 first of the baskets: there are A^i! different orders in which they could have 

 come out of the sack, and every one of these corresponds to the very same 

 inventory. Think next of the A% numbered balls which rest in the second 

 basket: they might have come out of the sack in A''2! different orders, without 

 changing the inventory. Think now of the contents of both of these baskets 

 at once. Each of the A'2! orders in which the second basketful may come 

 out of the sack may follow on any one of the A^i ! orders in which it is possible 

 for the first basketful to emerge. The product A^'i! A''2! is therefore the total 



