Generalized Method of Intervals Generalized Method of Intervals for Solving Inequalities: Let . Where k₁, k₂, … kn ϵ Z and a₁, a₂, …., an are fixed real numbers satisfying the condition. a₁ < a₂ < a₃ < … < a n₋₁ < an For solving F(x) > 0 (or) F(x) < 0, consider the Read more about Generalized Method of Intervals[…]
Multinomial Theorem Consider the equation x₁ + x₂ + … + xᵣ = n, where aᵢ ≤ xᵢ ≤ bᵢ; xᵢ ϵ 1; I = 1, 2, …, r. In order to find the number of solutions of the given equation satisfying the given condition. We observe that the number of solutions is the same Read more about Multinomial Theorem[…]
Properties of ⁿC r – III 5.. Proof: L.H.S = . . . . . . . Hence proved . 6. . 7. Maximum value of ⁿCr we can observe that in the list of ⁶C₀, ⁶C₁, ⁶C₂, ⁶C₃, ⁶C₄, ⁶C₅ and ⁶C₆the maximum value is ⁶C₃. Also, in the list of ⁷C₀, ⁷C₁, ⁷C₂, Read more about Properties of ⁿC r – III[…]
Properties of ⁿCr – II 3) ⁿCr + ⁿC(r-1) = (ⁿ⁺1)Cr L.H.S = ⁿ C r + ⁿ C (r -1) . . . . . . . . . Hence L.H.S = R. H. S ⁿCr + ⁿC(r-1) = ⁿ⁺1Cr 4. rⁿCr = n(ⁿ ⁻ ¹) C(r-1) L. H. S = rⁿCr . . Read more about Properties of ⁿCr – II[…]
Combination – Properties of ⁿCr Each of the different groups of selections, which can be made by taking some or all of a number of given things or objects at a time is called a combination. In combination, order of appearance of things is not taken into account Example: Three groups can be made with Read more about Combination – Properties of ⁿCr[…]
Permutations and Combinations – Division of Objects into Groups Division of items into groups of unequal size: Number of ways in which (m + n)! distinct items can be divided into two unequal groups containing m and n items is (m + n)! / (m!. n!). The number of ways in which (m + n Read more about Permutations and Combinations – Division of Objects into Groups[…]
Permutations Number of permutations of n different thing taken all at a time is n! Proof: By the multiplication rule, the number of ways of filling up the first, second, third … nth places together is n (n – 1) (n – 2) …. 2 x 1 = n!. Thus . . Factorial Zero: From Read more about Permutations[…]
Factorial of a Non – Negative Integer If n is a non-negative integer then factorial n is denoted by n! (or) n defined as follows (i) 0! = 1 (ii) If n > 0 then n! = n (n – 1)! The number of permutations of n dissimilar things taken all at a time is Read more about Factorial of a Non – Negative Integer[…]
Mean and Median Mean: The sum of all the observation is divided by the number of observations is called mean and it is denoted by x̅. (i) Mean of ungrouped (or) individual data. If x₁, x₂, …, xn are n observations, then mean x̅ (ii) Mean of grouped (or) continuous data. Whose corresponding frequencies are Read more about Mean and Median[…]
Functional Equations Resulting from Properties of Functions 1. Odd functions having symmetry of graph about the origin: f(x) + f(-x) = 0, x ϵ Df. 2. Even functions having symmetry of graph about the y – axis: f(x) = f(-x), x ϵ Df. 3. Symmetry of graph about the Point (a, 0): f(a – x) Read more about Functional Equations Resulting from Properties of Functions[…]