t into the bag, it does not matter in which order they have been selected. All permutations of these five specific fruits, (which are 5!), represent the same final bag selection. Therefore, the above count should be divided by 5! because of the 5! possible redundancies. The end result is, therefore, Q = 100!/[5!(100 - 5)!]. Most generally, the number of ways to pick up p unordered samples out of a set of n items is (1.9) The number CR, , which is also noted (p) or nCp or C(n, p) is called the binomial coefficient. 4 Since the factorial is expandable to a continuous function (see previous note), the binomial coeff- icient is most generally defined for any real/complex x, y numbers as Cx = I (x + T(y)[T(x-y+1)]. Beautiful plots of CX in the real x, y plane, and more on the very rich binomial 1.3 Combined, joint, and conditional probabilities A final example is going to show how we can use the binomial coefficient in probability analysis. Scooping jellybeans A candy jar contains 20 jellybeans, of which three quarters are green and one quarter is red. If one picks up ten jellybeans at random, what is the probability that all beans in the selection will be green? Answer: By definition, the number of ways to pick up 10 green beans out of 15 is C10. The total number of possible picks, i.e., a selection of 10 out of 20, is C20. Applying the probability definition in Eq. (1.4), with the event being "all picked beans are green," we obtain: p(all green) (1.10) 0.016. The result shows that the likelihood of getting only green jellybeans is relatively low (near 1.5%), even if 3/4 of the jar are of this type. Interestingly, if the jar had only one red jelly bean and 19 green ones, the event probability would be p(all green) = 0.5, as the reader should easily verify. It is also easy to show as an exercise that for a jar of one red jelly bean and N - 1 green jelly beans, the probability p(all green) becomes asymptotically close to unity as N reaches infinity, which is the expected result. Combined, joint, and conditional probabi