### Permutations and Combinations

goals
We are required to form different words with the help of letters of the word INTEGER. Let $m_1$ be the number of words in which $I$ and $N$ are never together and $m_2$ be the number of words which begin with $I$ and with $R$, then prove that $m_1/m_2 = 30$
Every composite number can be expressed as product of  powers of primes $N = 2^{a} \times 3^{b} \times 5^{c} \times$......where a, b, c are non-negative integers.
The total number of factors for N are $(a + 1) (b + 1) (c + 1) .....$
The sum of divisors $(s_{n}) = \left ( \frac{2^{a+1}-1}{2-1} \right )\left ( \frac{3^{b+1}-1}{3-1} \right )\left ( \frac{5^{c+1}-1}{5-1} \right )$
Find the number of $4$-letter words, that can be formed from the letters of the word $''ALLAHABA''$
Prove that: $^{2n}C_n = \dfrac{2^n \{1\cdot 3 \cdot 5 \cdot ....(2n - 1)\}}{n!}$
A round table conference is to be held between delegates of $20$ countries. In hoe many ways can they be seated if two particular delegates are:
a) Always together?
b) Never together?