## Permutations and Combinations

# Permutations

**Tips :**

- If a operation can be performed in ‘m’ different ways, following which a second operation can be performed in ‘n’ different ways then the two operations in succession can be performed in m × n different ways.
- If an operation can be performed in ‘m’ different ways and another operation, which is independent of the first operation, can be performed in ‘n’ different ways. Then either of ‘2’ operations can be performed in (m + n) ways.
- Permutation of things means arrangement of things. The word arrangement is used if order of things is taken into account.
- Number of permutation of ‘n’ different things taken all at a time is
^{n}P_{n}= n ! - The number of permutations of ‘n’ things, taken all at a time, out of which ‘P’ are alike and are of one type, q are alike and are of second type and rest are all different is

\tt \frac{n!}{P!q!} - The number of permutations of ‘n’ different things taken ‘r’ at a time when each thing may be repeated any number of times is n
^{r}.

**Tricks:**

- Number of permutations of ‘n’ different things, taken ‘r’ at a time when a particular thing is to be always included in each arrangement is r. \tt ^{n-1}P_{r-1}
- Number of permutations of ‘n’ different things, taken ‘r’ at a time when a particular thing is never taken in each arrangement is \tt ^{n-1}P_{r}
- Number of permutations of ‘n’ differnt things, taken all at a time when “m” specified things always come together, is m! × (n - m + 1)!
- Number of positive number not more than ‘P’ digits, if the digits 0, 1. ----- are used when repetitions of digits are allowed is n
^{P}– 1 where ‘n’ is number of digits including zero. - To arrange boys girls in a row alternately they should be equal in number or with different '1'. Otherwise it is not possible to arrange then alternately in a row.
- Sum of the number formed by taking all the given ‘n’ digits [excluding 0] is (sum of all ‘n’ digits) × (n – 1) ! × (111.....n times)
- When ‘n’ digits are given excluding zero then the sum of digits in units place of a ‘n’ digited number is (n – 1) ! × sum of numbers
- When ‘n’ digits are given excluding zero then the sum of digits in 100’s place of ‘n’ digited number is (n – 1) ! × sum of no.s × 100
- The number of circular permutations of ‘n’ dissimilar things taken ‘r’ at a time [clock and anti clock wises] is \tt\frac{ ^{n}P_{r}}{r}
- The number of circular permutations of ‘n’ things taken ‘r’ at a time in one direction is \tt\frac{ ^{n}P_{r}}{2r}
- The number of circular permutations of ‘n’ dissimilar things taken all at a time is (n – 1)!
- The number of ways in which ‘n’ dissimilar things can be arranged in a circular manner such that no one will have same neighbours in any two arrangements is \tt \frac{\left(n-1\right)!}{2}
- The number of ways in which m things and n things can be arranged in the form of garland so that no ‘2’ things of second kind come together is \tt \frac{\left(m-1\right)!^{m}P_{n}}{2}

In case all the 2nd type things should come together is \tt \frac{m!.n!}{2} - The number of ways in which exactly ‘r’ letters can be placed in wrongly addressed envelopes when ‘n’ letters are put in ‘n’ addressed envelopes is \tt ^{n}P_{r}\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+.....+\left(-1\right)^{r}\frac{1}{r!}\right]
- \tt \frac{^{n}P_{r}}{^{n}P_{r-1}}=\left(n-r+1\right)
- If r < s < n then \tt ^{n}P_{s} is divisible by \tt ^{n}P_{r}
- \tt 1.^{1}P_{1}+2.^{2}P_{2}+... n. ^{n}P_{n}=^{\left(n+1\right)}P_{\left(n+1\right)} - 1
- \tt If \frac{^{n}P_{r-1}}{a}=\frac{^{n}P_{r}}{b}=\frac{^{n}P_{r+1}}{c} then b
^{2}= a (b + c) - \tt ^{n}P_{r}+r. ^{n}P_{r-1}=^{\left(n+1\right)}P_{r}

### View the Topic in this video From 22:18 To 50:50

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1. The number of ways of arranging n distinct objects in a row taking r (0 ≤ r ≤ n) at a time is denoted by P(n, r) or ^{n}P_{r}.

i.e, \tt ^nP_r=\frac{n!}{\left(n-r\right)!}

2.** Properties of Permutation**

a) ^{n}P_{n} = n (n − 1) (n − 2)...1 = n!

b) \tt ^nP_0=\frac{n!}{n!}=1

c) ^{n}P_{1}=n

d) ^{n}P_{n-1} = n!

e) ^{n}P_{r}= n · ^{n-1}P_{r-1} = n(n − 1) · ^{n-2}P_{r-2}

=n(n−1)(n−2) · ^{n-3}P_{r-3}

f) ^{n-1}P_{r} + r · ^{n-1}P_{r-1} = ^{n}P_{r}

g) \tt \frac{^nP_r}{^nP_{r-1}}=n-r+1

3.** Important Results on Permutation**

a) The number of permutations of n different things taken r at a time, allowing repetitions is n^{r}.

b) The number of permutations of n different things taken all at a time is ^{n}P_{n} = n!.

c) The number of permutations of n things taken all at a time, in which p are alike of one kind, q are alike of second kind and r are alike of third kine and rest are different is \tt \frac{n!}{p!q!r!}.

d) Number of permutations of n different things taken r at a time, when a particular thing is to be included in each arrangement is r · ^{n-1}P_{r-1}

4. **Division into Groups**

a) The number of ways in which (m + n) different things can be divided into two groups which contain m and n things respectively =\tt \frac{\left(m+1\right)!}{m!n!}.

b) The number of ways in which mn different things can be divided equally it into m groups, if order of the group is not important is \tt \frac{\left(mn\right)!}{\left(n!\right)^{m}m!}.

c) If the order of the group is important, then number of ways of dividing mn different things equally into m distinct groups is \tt \frac{\left(mn\right)!}{\left(n!\right)^{m}}.

5. **Circular Permutation**

a) Number of circular permutations of n different things taken all at a time, when clockwise or anti-clockwise order is not different =\tt \frac{1}{2}\left(n-1\right)!.

b) Number of circular permutations of n different things taken r at a time, when clockwise or anti-clockwise orders are take as different is \tt \frac{^nP_r}{r}.

c) Number of circular permutations of n different things taken r at a time, when clockwise or anti-clockwise orders are not different is \tt \frac{^nP_r}{2r}.