**Applications of Permutation and Combination**

**Functional Applications**

(i) The number of all permutations (arrangements) of n different objects taken r at a time,

- When a particular object is to be always included in each arrangement is
^{n-1}C_{r-1}x r! - When a particular object is never taken in each arrangement is
^{n-1}C_{r}x r!

(ii) If the sets A has m elements and B has n elements, then

- The number of functions from A to B is n
^{m}. - The number of one-one functions from A to B is
^{n}P_{m}, m ≤ n. - The number of onto functions from A to B is \({{n}^{m}}\,-\,\left( \begin{matrix} n \\ 1 \\\end{matrix} \right){{\left( n-1 \right)}^{m}}+\left( \begin{matrix} n \\ 2 \\\end{matrix} \right){{\left( n-2 \right)}^{m}}-\,…\), m≤ n.
- The number of increasing (decreasing) functions from A to B is \(\left( \begin{align}& n \\ & m \\ \end{align} \right)\), m ≤ n.
- The number of non-decreasing (non-increasing) functions from A to B is \(\left( \begin{align}& m+n-1 \\ & \,\,\,\,\,\,m \\ \end{align} \right)\), m ≤ n.
- The number of bijections from A to B is n! if m = n.
- The number of bijections from A to A such that f(x) ≠ x, ” x ϵ A, is \(m!\left[ \frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-….+\frac{{{(-1)}^{m}}}{m!} \right]\).

**Example:** Let A = {x₁, x₂, x₃, x₄, x₅}, B = {y₁, y₂, y₃, y₄, y₅}. Then, the number of one-one mappings, from A to B such that f (x_{i}) ≠ y_{i}, i = 1, 2, …, 5 is

**Interpret:** Here, we use the formula, the number of bijections from A to A such that f (x) ≠ x, ” x ϵ A is \(m!\left[ \frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}-…..+\frac{{{(-1)}^{m}}}{m!} \right]\)

Here required number of ways \(=5!\left( \frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!} \right)\)

\(=120\left( \frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120} \right)\)

\(=120\left( \frac{60-20+5-1}{120} \right)=44\).

**Geometrical Applications:**

- Out of n non-concurrent and non-parallel straight lines the number of point of intersection are
^{n}C₂. - The number of straight lines passing through n points =
^{n}C₂. - The number of straight lines passing through n points out of which m are collinear
^{n}C₂ –^{m}C₂ + 1. - In a polygon, the total number of diagonals out of n points (no three points are collinear) =
^{n}C₂ – n = n(n-3)/ 2. - Number of triangles formed by joining n points is
^{n}C₃. - Number of triangles formed by joining n points out of which m are collinear are
^{n}C₃ –^{m}C₃. - The number of parallelogram in two systems of parallel lines (when 1
^{st}set contains m parallel lines and 2^{nd}set contains n parallel lines) =^{n}C₂ x^{m}C₂. - The number of rectangles of any size in a square of n x n is \(\sum\limits_{r\,\,=\,\,1}^{n}{{{r}^{3}}}\) and number of squares of any size is \(\sum\limits_{r\,\,=\,\,1}^{n}{{{n}^{2}}}\).