Ch.+11+-+Permutations+and+Combinations

v Solving counting problems using the fundamental counting principle v Determining using a variety of strategies, the number of permutations of n elements take r at a time v Solving counting problems when two or more elements are identical v Solving an equation that involves, Permutations and combinations. The fundamental counting principle can be used to determine the number of different arrangements. If one task can be performed in // a // ways, a second task in // b // ways, and a third task in // c // ways, then all three tasks can be arranged in // a×b×c // ways. Factorial notation is an abbreviation for products of successive positive integers. 5! = (5)(4)(3)(2)(1) ( // n // +1) = ( // n // +1)( // n // )( // n // -1)( // n // -1)( // n // -2)…(3)(2)(1)
 * == ** 11.1 Permutations ** ==
 * //__ Focus on… __//**
 * Key Ideas **

** 11.2 Combinations **
v Explaining the differences between a permutation and a combination v Determining the numbers of ways to select // r //elements from // n // different elements v Solving problems using the number of combinations of // n // different elements taken // r // at a time v Solving an equation that involves n C r notation.
 * //__ Focus on… __//**

A selection of objects in which order is not important is a combination. When determining the number of possibilities in a situation, if order matters, it is a permutation. If order does not matter, it is a combination. The number of combinations of n objects taken r at a time can be represented by n C r, where // n≥r // and // r // ≥0. A formula for n C r is n C r = n P r / r! or n C r = n! / ((n-r)!r!)
 * Key Ideas **

__ Focus on… __
v Relating the coefficients in the expansion of // (x+y) // // n ////, nϵN // , to Pascal’s triangle and to combinations v Expanding // (x+y) // // n ////, nϵN // , in a variety of ways, including the binomial theorem v Determining a specific term in the expansion of // (x+y) // // n //

Pascal’s triangle has many patterns. For example, each row begins and ends with 1. Each number in the interior of any row is the sum of the two numbers to its left and right in the row above. You can use Pascal’s triangle or combinations to determine the coefficients in the expansion of // (x+y) //// n //, where n is a natural number. You can use the binomial theorem to expand any binomial of the form // (x+y) //// n ////, nϵN //. You can determine any term in the expansion of // (x+y) //// n // using patterns without having to perform the entire expansion. The general term, // t //// k+1 // , =  Notes =
 * Key Ideas **
 * ==[[file:11.0 – The Fundamental Counting Principle K.pdf|11.0 – The Fundamental Counting Principle ]] ==
 * ==[[file:11.1– Permutations.pdf|11.1– Permutations]] ==
 * ==[[file:11.2– Combinations.pdf|11.2– Combinations]] ==