Topics
Sets and Relations
- Fundamental Concepts of Ordered Pairs and Relations
- Representation of a Set
- Intervals
- Classification of Sets
- Relations of Sets
Functions
- Domain and Range of a Function
- Types of Functions
- Representation of Function
- Representation of Function
- Fundamental Functions
- Algebra of Functions
- Composite Function
- Inverse Functions
- Some Special Functions
Complex Numbers 33
- Introduction of Complex Number
- Imaginary Number
- Concept of Complex Numbers
- Conjugate of a Complex Number
- Algebraic Operations of Complex Numbers
- Square Root of a Complex Number
- Solution of a Quadratic Equation in Complex Number System
- Cube Root of Unity
Sequences and Series
- Sequence, Series, and Progression
- Geometric Progression (G. P.)
- General Term Or the nth Term of a G.P.
- Sum to' n' Terms of a Geometric Progression
- Sum to' n' Terms of a Geometric Progression
- Recurring Decimals
- Harmonic Progression (H. P.)
- Types of Means
- Special Series (Sigma Notation)
Locus and Straight Line
- Locus
- Line
- Equations of Lines in Different Forms
- Equations of Line in Different Forms
Determinants
- Determinant of a Matrix
- Properties of Determinants
- Application of Determinants
- Determinant Method (Cramer’s Rule)
- Consistency of Three Linear Equations in Two Variables
- Area of a Triangle Using Determinants
- Collinearity of Three Points
Limits
- Definition of Limit of a Function
- Algebra of Limits
- Evaluation of Limits
- Direct Method
- Methods to Find Limit of Rational Function>Factorization Method
- Methods to Find Limit of Rational Function> Rationalization Method
- Limits of Exponential and Logarithmic Functions
Continuity
- Continuous and Discontinuous Functions
- Examples of Continuous Functions Whereever They Are Defined
Differentiation
- The Meaning of Rate of Change
- Definition of Derivative and Differentiability
- Derivative by the Method of First Principle
- Rules of Differentiation (Without Proof)
- Applications of Derivatives
Partition Values
- Partition Values
- Deciles
- Percentiles
- Relations Among Quartiles, Deciles and Percentiles
- Graphical Location of Partition Values
Measures of Dispersion
- Measures of Dispersion
- Quartiles and Range in Statistics
- Measures of Dispersion > Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Skewness
- Skewness
- Asymmetric Distribution (Positive Skewness)
- Asymmetric (Negative Skewness)
- Measures of Skewness
- Karl Pearson’S Coefficient of Skewness (Pearsonian Coefficient of Skewness)
- Features of Pearsonian Coefficient
- Bowley’s Coefficient of Skewness
Bivariate Frequency Distribution and Chi Square Statistic
- Bivariate Frequency Distribution
- Classification and Tabulation of Bivariate Data
- Marginal Frequency Distributions
- Conditional Frequency Distributions
- Categorical Variables
- Contingency Table
- Chi-Square Statistic ( χ2 )
Correlation
- Correlation
- Concept of Covariance
- Properties of Covariance
- Concept of Correlation Coefficient
- Scatter Diagram
- Interpretation of Value of Correlation Coefficient
Permutations and Combinations
- Fundamental Principles of Counting
- Concept of Addition Principle
- Concept of Multiplication Principle
- Concept of Factorial Function
- Permutations
- Circular Permutations
- Combination
Probability
- Concept of Probability
- Elementary Types of Events and Properties of Probability
- Elementary Properties of Probability
- Addition Theorem of Probability
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
Linear Inequations
- Linear Inequations
- Method of Solving a Linear Inequality
- Representation of Inequalities
- Graphical Solution of Linear Inequality of Two Variable
- Solution of System of Linear Inequalities in Two Variables
Commercial Mathematics
- Percentage
- Profit and Loss
- Simple and Compound Interest (Entrance Exam)
- Concept of Depreciation
- Partnership
- Shares
Estimated time: 7 minutes
Maharashtra State Board: Class 12
Formula: Combination
The number of combinations of n different things taken r at a time is
C(n, r) or \[^nC_r\] or \[\begin{pmatrix} n \\ r \end{pmatrix}\] i.e. \[^nC_r=\frac{n!}{r!(n-r)!},0\leq r\leq n\]
Maharashtra State Board: Class 12
Key Points: Combination
- \[^nC_{n-r}=^nC_r\mathrm{~for~}0\leq r\leq n\]
- \[{}^{n}C_{0}={}^{n}C_{n}=1\]
- \[{}^{n}C_{0}={}^{n}C_{n}=1\]
- If \[^nC_r={}^nC_s\], then either s = r or s = n - r.
- \[^nC_r=\frac{^nP_r}{r!}\]
- \[^nC_r+{}^nC_{r-1}={}^{n+1}C_r\]
- \[^nC_0+{}^nC_1+...+{}^nC_n=2^n\]
- \[^nC_0+^nC_2+^nC_4+...\] \[={}^{n}C_{1}+{}^{n}C_{3}+{}^{n}C_{5}+....=2^{(n-1)}\]
- \[^nC_r=\left(\frac{n}{r}\right)^{(n-1)}C_{(r-1)}=\left(\frac{n}{r}\right)\left(\frac{n-1}{r-1}\right)^{(n-2)}C_{(r-2)}=...\]
- \[^nC_r\] has maximum value, if
a. \[\mathrm{r=\frac{n}{2}}\], when n is even.
b. \[\mathbf{r}=\frac{\mathbf{n}-1}{2}\] or \[\frac{\mathrm{n}+1}{2}\], when n is odd.
Shaalaa.com | Theorem: `"^nC_r` + `"^nC_r-1`= `"^(n+1)C_r`
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