Binomial coefficients

The binomial distribution applies to random variables where there are only two possible outcomes (A or B) for each trial and where the outcome probability is constant over all n trials. If the probability of A occurring on any one trial is denoted as p and the number of occurrences of A is denoted as x, then the binomial coefficient is given by... 

The number of ways in which n bonds are arranged around an atom with average valence v in a crystal involves a binomial coefficient ... 

A table of binomial coefficients is given below (table IV). The numbers resulting from the successive partial differentiations, are given in table V, which can easily be extended with the use of the recursion formula... 

The notation employed in Eq. (5) corresponds to that defined in Section 2.10, namely, the binomial coefficients. Thus, in general the binomial coefficients in the polynomial expansion... 

Intensities can be calculated using the rule of binomial coefficients. The relative intensities in a simple multiplet (only one type of coupling neighbour) are as follows ... 

Figure 42 shows spectra of the simplest tetraorganotin compound, tetramethyltin. The upper spectrum was recorded with complete decoupling of all protons, the middle spectrum without. The result is a multiplet with 13 lines (n = 12), but if you work out the binomial coefficients for such a multiplet you will see that the outer two lines are too weak to be seen. The lower spectrum is the proton spectrum, which shows satellites due to two-bond tin-proton coupling to the tin-117 (inner lines) and tin-119 (outer lines) nuclei. 

When several magnetically equivalent nuclei are present in a radical, some of the multiplet lines appear at exactly the same field position, i.e., are degenerate , resulting in variations in component intensity. Equivalent spin-1/2 nuclei such as 1H, 19F, or 31P result in multiplets with intensities given by binomial coefficients (1 1 for one nucleus, 1 2 1 for two, 1 3 3 1 for three, 1 4 6 4 1 for four, etc.). One of the first aromatic organic radical anions studied by ESR spectroscopy was the naphthalene anion radical,1 the spectrum of which is shown in Figure 2.2. The spectrum consists of 25 lines, a quintet of quintets as expected for hyperfine coupling to two sets of four equivalent protons. 

For sets of spin- nuclei, the multiplet intensity ratios are simply the binomial coefficients found most easily from Pascal s triangle (Figure 2.11). 

Figure 2.11 Pascal s triangle for the determination of binomial coefficients.

Example V-2 gives a program to compute a binomial coefficient, z =... 

VERIFICATION OF A WHILE PROGRAM TO COMPUTE A BINOMIAL COEFFICIENT... 

Ghent, A.W. (1972). A method for exact testing of 2x2, 2x3, 3x3 and other contingency tables, employing binomiate coefficients. Am. Midland Naturalist. 88 15-27. 

The relative intensities of the lines within each multiplet will be in the ratio of the binomial coefficients (Table 5.9). Note that, in the case of higher multiplets, the outside components of multiplets are relatively weak and may be lost in the instrumental noise, e.g. a septet may appear as a quintet if the outer lines are not elearly visible. The intensity relationship is the first to be significantly distorted in non-ideal cases, but this does not lead to serious errors in speetral analysis. 

Let us illustrate the application of Boltzmann s formula for an elementary example isothermal mixing of ideal gases (Sidebar 5.10). For this purpose, consider a system of Na = Nb = 4 particles. For a particular partition n 4 — n of the four A-type particles between the VA and VB containers, the number of possible ways H of choosing n A-type particles and 4 — n B-type particles for the first container is given by the product of binomial coefficients... 

Carrying over the ideas of Section 8.5 (the derivation is formally the same as the derivation of the A portion of the NMR spectrum of an AX molecule), we see that the ESR spectrum in this case consists of a multiplet Of / +1 lines centered at gefieB0h l the spacing between adjacent lines is A, and the relative intensities are given by the binomial coefficients (8.86). For example, the ESR spectrum of C6H is shown in Fig. 8.13. 

Beveridge, D. L., 380 Binomial coefficients, 351,374 Biochemical applications of ESR, 380-381 of IR spectroscopy, 268 of NMR, 357, 363-364 of Raman spectroscopy, 270-271 Birge-Sponer extrapolation, 304-305 Blackbody radiation, 121-122 Bloch, F 328 Block-diagonal form, 15 Bohr (unit), 23 Bohr magneton, 51, 368 Bohr radius, 42 Bolometer, 260... 

It is understood that the binomial coefficient equals zero unless (r — n) is an integer between 0 and r inclusive. 

Note that this formula is valid for all integral values of provided one takes the binomial coefficient to be zero whenever nl < 0 or > n0. From this formula one derives, using a standard computational trick of classical probability theory,... 

The number, f2(n, NT). of different ways (trajectories, sequences, etc.) the walker can get to site n from the origin is given by the binomial coefficient... 

The sum in eq. (43) runs over all values of 0 < k < / I m, using ( n) = 1 when n 0 and (— ) =oo when n> 0. The formula (43) for matrix elements can be expressed in an alternative form that involves binomial coefficients instead of factorials (Altmann and Herzig (1994)) and this may be rather more useful for computational purposes. 

By some elementary manipulations of the binomial coefficients one arrives at... 

Eqs. (44) and (45) are especially convenient for practical applications because of the possible occurrence of vanishing binomial coefficients. 

The number of isotropic hyperfine lines from a particular nucleus depends on the nuclear spin, I, and the line multiplicity is 2/ + 1. For n equivalent nuclei, the EPR spectrum consists of 2nl + 1 lines whose relative intensities are given by binomial coefficients obtained in the expansion of (1 + x)n (Knowles et al., 1976). When nuclear hyperline interactions occur, Eq. (16.4) becomes... 

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