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In combinatorics, stars and bars (also called "sticks and stones", [1] "balls and bars", [2] and "dots and dividers" [3]) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve many simple counting problems , such as how many ways there are to put n indistinguishable balls into k distinguishable bins. [ 4 ]
The winner of the competition was Nicola Marschall's "Stars and Bars" flag. The "Stars and Bars" flag was only selected by the Congress of March 4, 1861, the day of the deadline. The first flag was produced in rush, due to the date having already been selected to host an official flag-raising ceremony, W. P. Miles credited the speedy completion ...
Stars and Bars, 1988 comedy starring Daniel Day-Lewis; Stars and Bars, 1917 silent film comedy directed by Victor Heerman; Stars and bars (combinatorics), a graphical method used to derive the formula for multiset coefficients and other combinatorial theorems; Stars and Bars, a 1984 novel by William Boyd; A song by Scottish group Goodbye Mr ...
The primary insignia was a number of bars or stars worn on the collar of the uniform coat or tunic. This was occasionally substituted for, or coupled with, shoulder straps . The secondary insignia were Austrian knots , elaborate patterns sewed on the lower sleeves with the number of lines indicating the rank.
The chart below is similar to the chart above, but instead of showing the formulas, it gives an intuitive understanding of their meaning using the familiar balls and boxes example. The rows represent the distinctness of the balls and boxes. The columns represent if multi-packs (more than one ball in one box), or empty boxes are allowed.
Of all the early operators of military aircraft, Germany was unusual in not using circular roundels. After evaluating several possible markings, including a black, red, and white checkerboard, a similarly coloured roundel, and black stripes, it chose a black 'iron cross' on a square white field, as it was already in use on various flags, and reflected Germany's heritage as the Holy Roman Empire.
[4] - PHS CC (28 Oct 2008), Stars and Bars, Uniform Accessories at Commissioned Corps-Rank and Insignia, ...when referring to commissioned officers, they traditionally are referred to by the equivalent U.S. Navy ranks (see table). The exception to this is that the proper titles and abbreviations of the Assistant Secretary for Health, Surgeon ...
Bar graphs can also be used for more complex comparisons of data with grouped (or "clustered") bar charts, and stacked bar charts. [5] In grouped (clustered) bar charts, for each categorical group there are two or more bars color-coded to represent a particular grouping. For example, a business owner with two stores might make a grouped bar ...