[A combinatorial approach to Question 11] In this question we consider binary words consisting only of the letters A and B. a Consider a binary word consisting of a + b letters, with A occuring a times and B occuring b times. Show that there are a+*Ca = a+*C» permutations of such a word. b How many possible permutations are there of a binary word with 2n letters, if A and B both occur n times? © Bill Pender et al. 2019 Photocopying is restricted under law and this material must not be transferred to another party Cambridge University Pr Extension 1 Year 11 ISBN 978-1-108-46907-4 aths Stage 6 15D Identities in Pascal's triangle c A word with 2n letters may be split down the middle into two words of n letters. Consider the example where two As fall in the first n-letter word. i How many arrangements are there of the first n-letter binary word with two As? ii How many arrangements are there of the second n-letter binary word with n - 2 of the As and 2 Bs? iii How many arrangements are there of a ten-letter binary word with two As in the first half and two Bs in the second half? Hence prove that d + ("C2n)? ("Co)² + ("C1)² + ("C2)² + ("C3)² + = 2"Cn- + . ..
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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