Permutations and Combinations
2. Fundamental Principle of Counting
2.1. Factorial Notation
Factorial is represented by the symbol ‘!’, n! =1×2×3×4….(n – 1)×n
It is always useful to have a simple notation for big products like 1×2×3×4×…..×99×100 = 100!.
For example 4! = 4.3.2.1 = 24, 5! = 5.4.3.2.1 = 120,
Note that 0! = 1.
Example 01: If \(1 \times 3 \times 5 \times 7 \times ........ \times 99 = \cfrac{{^{100}{{\rm{P}}_{50}}}}{k}\), then the value of \(k\) is:
The number \(1 \times 3 \times 5 \times 7 \times .... \times 99\) can be written as
\(\cfrac{{1 \times 2 \times 3 \times 4 \times .... \times 100}}{{2 \times 4 \times 6 \times .... \times 100}}\)
\( = \cfrac{{100!}}{{{2^{50}} \times 50!}} = \cfrac{{^{100}{{\rm{P}}_{50}}}}{{{2^{50}}}}\)
\(\cfrac{{1 \times 2 \times 3 \times 4 \times .... \times 100}}{{2 \times 4 \times 6 \times .... \times 100}}\)
\( = \cfrac{{100!}}{{{2^{50}} \times 50!}} = \cfrac{{^{100}{{\rm{P}}_{50}}}}{{{2^{50}}}}\)