# Permutations And Combinations class 11 Notes Mathematics

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## Permutations And Combinations class 11 Notes Mathematics

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CBSE Class 11 Mathematics
Revision Notes
Chapter-7
Permutations and Combinations

1. Fundamental Principle of Counting
2. Permutations
3. Combinations

Fundamental Principle of Counting

• Addition Law: If there are two operations such that such that they can be performed independently in $m$ and $n$ ways respectively, then either of the two operations can be performed in $(m+n)$ ways.
• Multiplication: If one operation can be performed in $m$ ways and if corresponding to each of the $m$ ways of performing this operation, there are $n$ ways of performing a second operation, then the number of ways of performing two operations together in $m \times n$.
• Factorial Notation: The continued product of first $n$ natural numbers is called the ‘$n$ factorial’ and is denoted by $n!$.
• $0! = 1$
• Permutations: The number of permutations of n different things taken r at a time, where repetition is not allowed, is denoted by $^{n}{{P}_{r}}$ and is given by $^{n}{{P}_{r}}=\frac{n!}{(n-r)!}$

$\text{where }0\text{ }\le \text{ r }\le \text{ n}.$

$\text{n}!\text{ }=\text{ 1 }\times \text{ 2 }\times \text{ 3 }\times \text{ }...\times \text{n}$

${\text{n}}!{\text{ }} = \;{\text{n }} \times {\text{ }}\left( {{\text{n }}-{\text{ 1}}} \right){\text{ }}!$

• The number of permutations of n different things, taken r at a time, where repeatition is allowed, is ${{n}^{r}}$.
• The number of permutations of  n objects taken all at a time, where  ${{\text{p}}_{1}}$ object are of first kind, ${{\text{p}}_{2}}^{{}}$ objects are of the second kind, …,${{\text{p}}_{k}}$ objects are of the ${{\text{k}}^{th}}$ kind and rest, if any, are all different is $\frac{n!}{{{p}_{1}}!{{p}_{2}}!..{{p}_{k}}!}.$

Combinations:

• The number of combinations of n different things taken r at a time, denoted by $^{n}{{C}_{r}}$ is given by $^{n}{{C}_{r}}=\frac{n!}{r!(n-r)!},o\le r\le n.$
• $^n{{\rm{C}}_0} = 1$
• $^n{{\rm{C}}_n} = 1$
• $^n{{\rm{C}}_r}{ = ^n}{{\rm{C}}_{n - r}}$
• $^n{{\rm{C}}_r}{ + ^n}{{\rm{C}}_{r - 1}}{ = ^{n + 1}}{{\rm{C}}_r}$
• $^n{{\rm{C}}_r} = {n \over r}{.^{n - 1}}{{\rm{C}}_{r - 1}}$
• $n{.^{n - 1}}{{\rm{C}}_{r - 1}} = {\left( {n - r + 1} \right)^n}{{\rm{C}}_{r - 1}}$
• Division into Groups: The number of ways $m+n$ things can be divided into two groups containing $m$ and $n$ things respectively = $^{m + n}{{\rm{C}}_m} = {{\left( {m + n} \right)!} \over {m!.n!}}$