Class 11 Mathematics:
Permutations, Combinations and Binomial theorem Demystified
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Welcome to Permutations and Combinations. Permutations and Combinations form an integral part of Algebra and is also used extensively in Probability. I will be giving you an overview of this chapter and the important topics that you should be knowing.
A good knowledge of Permutations and Combinations helps in answering SAT subject level exams and the GRE QUANT exams. Each problem has a different approach here and there is no standard formula applying to every problem.You can contact me for online classes if you need further help. To start with you will need to understand the factorial notation and how to solve problems using factorial. Next, you'll need to learn about the Fundamental Principle of Counting.
This is basically the principal concept used in every problem in this chapter. A permutation is basically an arrangement of objects. The number of permutations of n different objects, taken r at a time is denoted by P(n,r).
You start with the formula for permutations and use it in problem solving. An interesting case is when you have to find the number of permutations when two or more objects are together. In this case, you take these as one object and find the number of permutations of the remaining objects. Finally you have to remember to permute these objects also.
The next important point is to find the number of permutations when all the objects are not different. That is, you find the number of permutations of n objects taken all together, when p objects are alike of one kind, q of them are alike of another kind and r of them are alike of a third kind. These type of problems are used while considering the arrangements of alphabets.
You'll then proceed to learn about Circular Permutations. The number of permutations of n objects around a circular table is (n-1)! The number of ways of arranging n persons around a round table so that no person has the same two neighbours is (n-1)!/2. This is because the clockwise and anticlockwise arrangements are not different.
Moving on to Combinations, keep in mind that a combination is a selection. Learn how to find the combinations of n different objects taken r at a time. Again, there are a number of properties in Combinations which you'll be learning.
You'll also learn how to solve mixed problems which have both Permutations and Combinations. These problems are slightly tricky requiring an in depth understanding. Feel free to join my online classes for Permutations and Combinations.
You'll learn formulas, Concepts and problem solving skills.
Coming to Binomial Theorem, you need to have some knowledge of Combinations, before starting this chapter. Basically, the Binomial Theorem helps you to expand powers of (a+b).The coefficients of the powers of a and b are called Binomial coefficients.
Note, that the formula to evaluate powers of (a-b) is the same as a+b, only we replace b by -b. You can use this theorem to find the number of terms of (a+b)^n. You'll then learn the formula to find the general term of (a+b)^n.
This formula also helps you to determine the middle term of (a+b)^n. the constant term or the coefficient of any power of a or b. This is a fairly easy topic. However, if you need further help, you can join my online classes.
MIDDLE TERM OF A BINOMIAL EXPANSION
You will learn how to calculate the middle term/terms of a Binomial expansion. Note that, when n is even, there are n+1 terms, hence one middle term. If n is odd, there are even number of terms. Hence there are 2 middle terms. You can view the formula and a solved problem by clicking on this link.
HOW TO FIND THE TERM INDEPENDENT OF X
2 types of problems are discussed here. Firstly, we learn to find the term independent of x, using the formula for the r th term of a binomial expansion. Then you learn how to find the r th term from the end in any binomial expansion. You will understand that it is as simple as finding the r th term from the beginning, that is, the rth term.
Watch this video to learn how to expand using Binomial Theorem. A number of useful formulas are mentioned in this video. Lean how to find the number of terms of a Binomial expansion.
GENERAL TERM OF A BINOMIAL EXPANSION
Learn how to find the general term of a Binomial expansion. Notice how you can use this formula to find any term of an expansion.