## Permutations and Combinations

# Combinations

**Tips :**

- \tt ^{n}C_{o}+^{n}C_{1}+^{n}C_{2}+....+^{n}C_{n}=2^{n}
- \tt ^{\left(2n+1\right)}C_{o}+^{\left(2n+1\right)}C_{1}+....+^{\left(2n+1\right)}C_{n}=2^{2n}
- \tt ^{n}C_{r}+^{n}C_{r-1}=^{\left(n+1\right)}C_{r}
- \tt \frac{^{n}C_{r}}{^{n}C_{r-1}}=\frac{n-r+1}{r}
- If \tt ^{n}C_{r} is greatest then \tt r=\frac{n}{2} when ‘n’ is even \tt r=\frac{n-1}{2}or\frac{n+1}{2} when ‘n’ is odd
- \tt ^{m}C_{o}.^{n}C_{r}+^{m}C_{1}.^{n}C_{r-1}+.......+^{m}C_{r}.^{n}C_{o}=^{\left(m+n\right)}C_{r}
- ‘n’ different objects are in a row. The number of ways of selecting ‘r’ objects at a time so that no ‘2’ of these ‘r’ objects are consecutive is \tt ^{\left(n-r+1\right)}C_{r}
- Number of rectangles on a chess board (including squares) are 1296
- Number of squares on a chess board (exclusively squares) are 204
- Number of rectangles on a chess board which are not squares are 1092.

**Tricks :**

- The number of ways of answering one or more of ‘n’ questions is 2
^{n}- 1 - The number of ways of answering one or more of ‘n’ questions when each question have an alternative is 3
^{n}- 1 - The number of ways of answering all of ‘n’ questions when each question have an alternative is 2
^{n}. - If ‘n’ points are on the circumference of a circle are given then no of straight lines are
^{n}C_{2}, triangles are^{n}C_{3}. - If a polygon has ‘n’ sides then the no of diagonals are \tt \frac{n\left(n-3\right)}{2}
- In a plane there are ‘n’ points and no ‘3’ points are collinear except ‘k’ points which lie on a line. Then number of straight lines are “
^{n}C_{2}–^{k}C_{2}+1” and no of triangles are “^{n}C_{3}–^{k}C_{3}” - If a set of ‘m’ parallel lines are intersected by another set of ‘n’ parallel lines then the number of parallelograms that can be formed are “
^{m}C_{2}×^{n}C_{2}“ - If N = \tt p_1^{\alpha_{1}}.p_2^{\alpha_{2}}.p_3^{\alpha_{3}}....... p_k^{\alpha_{k}} where P
_{1}, P_{2}, P_{3}...... P_{k}are different primes and α_{1}, α_{2}, α_{3}- - - - α_{k}are natural numbers then total numbers of divisions of ‘N’ including ‘I’ and ‘N’ is (α_{1}+ 1) (α_{2}+ 1) (α_{3}+ 1)...... (α_{k}+ 1) - The sum of all divisions is \tt \left[\frac{p_1^{\alpha_{1}+1}-1}{p_{1}-1}\right]\left[\frac{p_2^{\alpha_{2}+1}-1}{p_{2}-1}\right]........ \left[\frac{p_k^{\alpha_{k}+1}-1}{p_{k}-1}\right]
- 10.The number of ways in which ‘N’ can be resolved as a product of two factors is \tt \frac{\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right)......\left(\alpha_{k}+1\right)}{2} [ if N is not a perfect square] \tt \frac{\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right).......\left(\alpha_{k}+1\right)+1}{2} [if ‘N’ is a perfect square]
- The number of ways in which “N” can be resolved into two factors which are relative prime = 2
^{r-1}, where ‘r’ is the number of distinct primes - For every number ‘N’ , ‘I’ and itself are always divisors these two are called trivial divisors and other are non-trivial.
- The number of ways of selecting ‘3’ natural numbers out of 2n consecutive natural numbers such that they are in A.P is 2.
^{n}C_{2} - The number of ways of selecting ‘3’ natural numbers out of (2n+1) consecutive natural numbers such that they are in A.P is
^{n}C_{2}+^{(n+1)}C_{2} - Selecting any ‘3’ numbers which are in A.P is equal to selecting any two numbers whose sum is even.
- Number of selections of ‘r’ things from ‘n’ things when ‘P’ particular things are not together in any selection is
^{n}C_{r}-^{n-p}C_{r-p} - Number of selections of ‘r’ consecutive things out of ‘n’ different things in a row = n – r + 1
- Number of selections of ‘r’ consecutive things out of ‘n’ different things around a circle is ‘n’ when r < n is ‘I’ when r = n
- Number of selections of ‘r’ things (r < n) out of ‘n’ identical things is ‘I’.
- Number of selections of none or more things out of ‘n’ identical things is “n + 1”.
- Number of selections of one or more things out of ‘n’ identical things is ‘n’.
- ‘n’ straight lines are drown in the plane such that no ‘2’ lines are parallel and no ‘3’ lines are concurrent. Then the number of parts into which these lines divide the plane is equal to \tt 1+\frac{n\left(n+1\right)}{2}
- The number of triangles whose angular points are at the angular points of a given polygon of ‘n’ sides, but none of whose sides are the sides of the polygon is \tt \frac{1}{6}n\left(n-4\right)\left(n-5\right).
- There are ‘n’ straight lines in a plane , no two of which are parallel and no three passes though the same point. Their points of intersection are joined. Then the number of fresh lines thus introduced is \tt \frac{1}{8}n\left(n-1\right)\left(n-2\right)\left(n-3\right).
- Number of points of intersections of diagonals of a polygon of ‘n’ sides which lie completely inside the polygon are “
^{n}C_{4}”.

### View the Topic in this video From 51:00 To 57:20

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1. The number of combinations of n different things taken r at a time is

\tt C\left(n,r\right)or \ ^nC_r \ or \left(\begin{array}{c}n\\ r\end{array}\right)

\tt ^nC_r = \frac{n!}{r!\left(n-r\right)!},\ 0\leq r \leq n

2. **Properties of Combination**

a) ^{n}C_{0} = ^{n}C_{n} = 1

b) ^{n}C_{1} = n

c) If ^{n}C_{r} = ^{n}C_{p}, then either r = p or r + p = n

d) ^{n}C_{r} = \tt \frac{^nP_r}{r!}

e) ^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r}

f) ^{n}C_{0} + ^{n}C_{1} + ^{n}C_{2} + ... + ^{n}C_{n} = 2^{n}

g) ^{n}C_{0} + ^{n}C_{2} + ^{n}C_{4} + ... = ^{n}C_{1} + ^{n}C_{3} + ... = 2^{n-1}

h) \tt \frac{n}{r}\ ^{n-1}C_{r-1}=\frac{n}{r}\frac{\left(n-1\right)}{\left(r-1\right)}\ ^{n-1}C_{r-2}

i) ^{2n+1}C_{0} + ^{2n+1}C_{1} + ^{2n+1}C_{2} + ... + ^{2n+1}C_{n} = 2^{2n}

j) ^{n}C_{n} + ^{n+1}C_{n} + ^{n+2}C_{n} + ... + ^{2n-1}C_{n} = ^{2n}C_{n+1}

3. **Important Results on Combination**

a) The number of combinations of n different things taken r at a time allowing repetitions is ^{n+r-1}C_{r}.

b) The number of ways of dividing n identical things among r persons such that each one gets atleast one is ^{n-1}C_{r-1}.

c) The total number of combinations of n different objects taken r at a time in which

i) m particular objects are excluded = ^{n-m}C_{r}.

ii) m particular objects are included = ^{n-m}C_{r-m}.

d) The total number of ways of dividing n identical items among r persons, each one of whom can receive 0, 1, 2 or more items (≤ n) is ^{n+r-1}C_{r-1}.

e) The total number of ways of selection of some or all of n things at a time is ^{n}C_{1} + ^{n}C_{2} + ... + ^{n}C_{n} = 2^{n} − 1.

4. **Arrangements**

a) The number of ways in which m (one type of different things) and n (another type of different things) can be arranged in a row so that all the second type of things come together is n !(m + 1)!.

b) The number of ways in which m (one type of different things) and n (another type of different things) can be arranged in row so that no two things of the same type come together is 2 × m! n!

c) The number of ways in which m (one type of different things) and n (another type of different things) (m ≥ n), can be arranged in a circle so that no two things of second type come together (m − 1)! ^{m}P_{n} and when things of second type come together = m! n!

d) The number of ways in which m things of one type and n things of another type (all different) can be arranged in the form of a garland so that all the second type of things come together, is \tt \frac{m!n!}{2} and if no two things of second type come together, is \tt \frac{\left(m-1\right)!^mP_n}{2}

5. **Selection**

a) The total number of ways in which it is possible to make a selection by taking some or all the given n different objects is

^{n}C_{1} + ^{n}C_{2} + ... + ^{n}C_{n} = 2 ^{n }− 1