If 2n+1 P_n-1 : 2n-1 P_n= 3 : 5, find n.

In how many ways can the letters of the word PERMUTATIONS be arranged if vowels are all together.

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if atleast 5 women have to be included in a committee? In how many of these committees the women are in majority?

Prove by the principle of mathematical induction that for every natural number n, 3^{2n + 2} - 8ⁿ - 9 is divisible by 8.

1 + 5 + 9 + ... + (4n - 3) = n(2n -1), \(\forall\)n \(\in\)N.