210 APPENDIX II—INTENSIVE SEGREGATION. 
digesting the same, so that, in each district alike, one in a million could 
survive in this way though the crop of leaves should fail. 
(4) Suppose that there are, through diversity of adaptations of this 
kind to products of the environment, ten different kinds of accessible 
forms of food, on each kind of which one in a million of the individuals 
of each district might feed if driven by necessity. 
(5) Now suppose the same necessity should occur in each district 
through the destruction of the leaves on which they habitually feed, 
and that there are accordingly in each district a hundred survivors 
able to maintain themselves on other kinds of food. 
Under such circumstances (the correspondences of which we have in 
our supposition made much more exact than the actual deviations from 
a mean ever present), even under such circumstances of completely 
parallel variation, what is the probability that in each of the separate 
districts the few that would meet with other individuals and have an 
opportunity to propagate the species would be similarly endowed and 
similarly related to the environment? 
In order to still further simplify the problem, let us assume that in 
the case of each kind in each district the probability that it will suc- 
ceed in propagating is exactly balanced by the probability that it will 
fail. The probability, then, that any given number, a, of the ten kinds 
in a given district will succeed is found by estimating the number of 
ways in which a things can be taken out of 10 things, and dividing 
this number by the tenth power of 2, that is, by 1024. This is com- 
pletely parallel to the number of ways in which ten pennies can be 
arranged as to head and tail, each penny representing one form of varia- 
tion, andits lying head-up indicating successin propagating. In 1024 
experiments the probability is— 
That o will succeed........ rtime | That 6 will succeed.........210 times 
That’ 1 will succeedi.\.).).4/5).. 10 times | That 7 will succeed........ 120 times 
That 2 will succeed ........ 45 times | That 8 will succeed......... 45 times 
‘bhat~3 willisucceed? 7o77..1 5, 120 times | That 9 will succeed........ 10 times 
That 4 will succeed... ...... 210 times | That ro will succeed........ 1 time 
That 5 will/succeéd........ 252 times 

These figures are found in the eleventh line of what is knownas the 
‘Table of the Binomial Coefficients,’’ or the ‘‘ Arithmetical Triangle.’’* 
And so in the case of any number of objects, the number of combina- 
tions that may be made with m objects is found in the m + 1th line of 
the arithmetical triangle classified according as there are 0, I, 2, 3, or 
more objects in each combination. The whole number of combina- 
tions may also be found by calculating the nth power of 2. 

* See ‘‘Principles of Science,’ by W_ S. Jevons. 
