Permutations and Combinations

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?