830 BEGUELIN. 



Here there are only 13 — 8, that is, 5 complete files; and 

 proceeding as in Art. 605, we find that the whole number of asso- 

 5x6x7x8x9 



ciations IS 



1x2x3x4x5 



In this way we arrive in fact at the value which we quoted 

 for E{7i, r) in Art. 602. 



608. The method which we have here briefly exemplified is 

 used by Begnelin in discussing all the parts of the problem. 

 He does not however employ letters as we have done ; he supposes 

 a series of medals of the Roman emperors, and so instead of 

 a, h, c,...he uses Augustus^ Tiberius, Caligula, ... 



609. It may be useful to state the results which are obtained 

 when there are n tickets of which 5 are drawn. 



In the following table the first column indicates the form, the 

 second the number of cases of that form according to Euler's 

 conception, and the third the number according to John Ber- 

 noulli's conception. 



Sequence of 5, n — 4, n. 



Sequence of 4, {n — 5) {n — 4), n{n — Q). 



Sequence of 3 



combined with (n-5) {n — 4<), n{n-6). 



a sequence of 2, 



Sequence of 3, 



and the other (n — 6) {n— o) [n — 4) n {n — 7) {n — 6) 



numbers not 1.2 ' 1.2 



in sequence, 



Two sequences {n — 6) {n— 5) {n — 4) n{n — 7) (n — 6) 



of 2, 1.2 ' 172 • 



Single sequence {n-7) (w-6) (n-5) (m-4) n {n-8) {n-7) (w-6) 

 of 2, 1.2.3 ' 17273 • 



No sequence, see Art. 602. 



