Permutation Formula and Facts

1. Factorial Notation:

Let n be a positive integer. Then, factorial n, denoted n! is defined as:

n! = n(n - 1)(n - 2) ... 3.2.1

Exmaple: 5! = 5*4*3*2*1 = 120
Note: Please remember 0! = 1

2. Permutation:
The different arrangements of a given number of things by taking some or all at a time, are called permutations.

Example:
i. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).

ii. All permutations made with the letters a, b, c taking all at a time are:
( abc, acb, bac, bca, cab, cba)

3. Number of Permutation:
Number of all permutations of n things, taken r at a time, is given by:

\begin{aligned}
^n P_r = n(n-1)(n-2).....(n-r+1) \\
= \frac{n!}{(n-r)!} \\

Example: \\
^4P_2 = 4 * 3 = 12 \\
^6P_2 = 6 * 5 = 30 \\

\end{aligned}

4. An Important Result:

If there are n subjects of which p1 are alike of one kind. p2 are alike of another kind. p3 are alike of third kind and so on and pr are alike of rth kind,
such that (p1 + p2 + ... pr) = n.

Then, number of permutations of these n objects is =

\begin{aligned}
\frac{n!}{(p_1!)(p_2!)(p_3!)....(p_r!)}
\end{aligned}