Revision: Permutations and Combination Maths HSC Science (General) 11th Standard Maharashtra State Board

If n is a positive integer, then the continued product of the first n natural numbers is called the factorial n, to be denoted by n! or |n|.

i.e., n! = n(n − 1)(n − 2) … 3 × 2 × 1

• 0! = 1
• n! = n(n − 1)!
• If x! = y! ⇒ x = y (for x, y ≥ 0)

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. \[^nC_r=\frac{n!}{r!(n-r)!},0\leq r\leq n\]

If two mutually exclusive operations can be performed in ‘m’ ways and ‘n’ ways respectively, then the number of ways in which either can occur is (m + n) ways.

If one operation can be performed in m different ways and another can be performed in n different ways, then the two operations taken together can be performed in (m × n) ways.

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!
• \[^nP_r\] = \[\mathrm{^{n-1}P_r+r.^{n-1}P_{r-1}}\]

1. The number of circular permutations of different objects = (n − 1)!.
2. If the clockwise and the counterclockwise orders are not distinguished, then the number of ways = \[\frac{1}{2}\left(n-1\right)!\].
3. The number of ways in which n objects of which p are alike, can be arranged in a circular order is \[\frac{(n-1)!}{p!}\].
4. Number of circular permutations of n different objects taken r at a time.
   a. When clockwise and anticlockwise orders are taken as different = \[\frac{^nP_r}{r}\].
   b. When clockwise and anticlockwise orders are not to be considered different, is = \[\frac{^nP_r}{2r}\].

• \[^nC_{n-r}=^nC_r\mathrm{~for~}0\leq r\leq n\]
• \[{}^{n}C_{0}={}^{n}C_{n}=1\]
• \[{}^{n}C_{0}={}^{n}C_{n}=1\]
• 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.