It is very easy to calculate binominal coefficients with the "calculating marble run", which is shown in the exhibition ix-quadrat of the Centre for Mathematics at the TU München (Director: Prof. Richter-Gebert) (PDF file). In nuclear magnetic resonance (NMR) spectoscopy the signal intensities i often follow such a binominal distribution. This shall be demonstrated with the nitrogen signal of the ammonium ion.
The ammonium ion (NH4+) as well as the methane (CH4) molecule have a tetrahedral structure. In case of the ammonium ion there is one central nitrogen atom bonded to four hydrogen atoms. Certain magnetic characteristics of hydrogen, carbon or nitrogen can be measured with NMR spectroscopy. The result of such a measurement is a spectrum, where different atomic nuclei of one molecule give rise to signals with different frequencies.
The spectrum of the tetrahedral ammonium ion consists, roughly speaking, of one signal at the characteristic frequency of nitrogen and a signal at the characteristic frequency of hydrogen. (In fact only the rare nitrogen isotope 15N delivers an NMR signal.) Taking a closer look at the signals one can observe that they consist of several individual components. The enlarged image of the nitrogen signal of the ammonium ion shows that it consists of five individual components with different intensities.
The interaction between the nitrogen atom and the four bounded hydrogen atoms causes this signal structure. The so called nuclear spins of the hydrogen atoms are either oriented to the top or to the bottom. The exact frequency of the individual components of the nitrogen signal depends on the number of hydrogen nuclear spins which are oriented to the top.
The relative size of the five individual components corresponds to the respective amount of combination possibilities to find a certain number of hydrogen nuclear spins which point to the top. There exists, for example, just one combination, where all four spins point to the top. But there are four combinations, where three spins are oriented to the top and six combinations when two spins are oriented to the top. (see picture).
The resulting size of the individual components of the nitrogen NMR signal correlates just to the binominal coefficients, which are delivered by the calculating marbel run: 1:4:6:4:1.
Dr. R. Marx and Prof. S. Glaser (2005)