Formulae [1]
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. \[^nC_r=\frac{n!}{r!(n-r)!},0\leq r\leq n\]
Key Points
Key points: Permutations
When all the objects are distinct
- \[^nP_r\mathrm{~or~}P(n,r).\mathrm{~i.e.~}P\left(n,r\right)={}^nP_r=\frac{n!}{\left(n-r\right)!},0\leq r\leq n\]
- Repetition Allowed: nn
- Objects Together: 2(n − 1)!
- Specified Object Included: \[r\times^{(n-1)}P_{(r-1)}\]
When some objects are identical
- Identical objects (p same, rest distinct): \[\frac{n!}{p!}\]
- Multiple identical objects: \[\frac{n!}{p_{1}!p_{2}!\ldots p_{k}!}\]
Special Cases:
- \[^nP_n\] = \[\mathrm{n!}\]
- \[{}^{\mathrm{n}}\mathrm{P}_{0}=1\]
- \[^nP_1\] = n
- \[{}^{n}P_{n-1}\]= n!
- \[^nP_r\] = \[\mathrm{^{n-1}P_r+r.^{n-1}P_{r-1}}\]
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.
