Permutations and Combinations - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter-7
Permutations and Combinations

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

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

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

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

• The number of permutations of n different things, taken r at a time, where repeatition is allowed, is nr$n^r$.

• The number of permutations of  n objects taken all at a time, where  p1${\text{p}}_1$ objectare of first kind, p2${{\text{p}}_2}^{}$ objects are of the second kind, ...,pk${\text{p}}_k$ objects are of the kth${\text{k}}^{th}$ kind and rest, if any, are all different is n!p1!p2!..pk!.$\frac{n!}{p_1!p_2!..p_k!}.$

Combinations:

• The number of combinations of n different things taken r at a time, denoted by nCr${}^n{C}_r$ is given by nCr=n!r!(n−r)!,o≤r≤n.${}^n{C}_r=\frac{n!}{r!(n-r)!},o\le r\le n.$
• nC0=1${}^n{\mathrm{C}}_0=1$
• nCn=1${}^n{\mathrm{C}}_n=1$
• nCr=nCn−r${}^n{\mathrm{C}}_r{=}^n{\mathrm{C}}_{n-r}$
• nCr+nCr−1=n+1Cr${}^n{\mathrm{C}}_r{+}^n{\mathrm{C}}_{r-1}{=}^{n+1}{\mathrm{C}}_r$
• nCr=nr.n−1Cr−1${}^n{\mathrm{C}}_r=\frac{n}{r}{.}^{n-1}{\mathrm{C}}_{r-1}$
• n.n−1Cr−1=(n−r+1)nCr−1$n{.}^{n-1}{\mathrm{C}}_{r-1}={\left(n-r+1\right)}^n{\mathrm{C}}_{r-1}$
• Division into Groups: The number of ways m+n$m+n$ things can be divided into two groups containing m$m$ and n$n$ things respectively = m+nCm=(m+n)!m!.n!