JOHN BERNOULLI. 327 



that on his conception of a sequence these formulae will hold if we 

 change n into n — 1. He does not demonstrate this statement, 

 so that we cannot say how he obtained it. 



It may be established by induction in the following way. Let 

 ^ (n, r) denote the number of ways in which we can take r tickets 

 out of n, free from any sequence, on Euler's conception of a se- 

 quence. Let B [n, r) denote the corresponding number on John 

 BernoulH's conception. Then we have given 



^, . (n—r-{-l)(n — r)...(n — 2r-\-2) 

 E{n, r) = ^ ^ ^ ^ ^ , 



and we have to shew that 



^ , . n(n — T — 1) ... (71— 2r + 1) 

 B in, r) = — ^ '- ^ . 



For these must be the values of E (w, r) and B [n, r) in order 

 that the appropriate chances may be obtained, by dividing by the 

 total number of cases. Now the following relation will hold : 



E{n,r)=B{n,r) + B{7i-l, r -1) -E{n-2, r-l). 



The truth of this relation will be seen by taking an example. 

 Suppose n is 10, and r is 3. Now every case which occurs in 

 the total B [n, r) will occur among the total E {n, r) ; but some 

 which do not occur in B{n,r) will occur in E{n,r), and these 

 must be added. These cases which are to be added are such as 



(10, 1, 3) (10, 1, 4) (10, 1, 8). We must then examine by what 



general law we can obtain these cases. We should form all the 

 binary combinations of the numbers 1 . 2, ... 9 which contain no 

 Bernoullian sequence, and which do contain 1. 



And generally we should want all the combinations r — 1 at a 

 time which can be made from the first n—1 numbers, so as to con- 

 tain no Bernoullian sequence, and to contain 1 as one of the num- 

 bers. It might at first appear that B (n — 1, r — V)—B(7i—2, r — 1) 

 would be the number of such combinations ; but a little con- 

 sideration will shew that it is B {n — 1, r — 1) — E {n — 2, 7* — 1), as 

 we have given it above. 



Thus having established the relation, and found the value of 

 B {ti, 1) independently we can infer in succession the values of 

 B (n, 2), B {n, 3), and so on. 



