Find the number of strings that can be made using all letters of the word THING. If these words are written as in a dictionary, what will be the 85^th string?

Find the sum of all 4-digit numbers that can be formed using digits 1, 2, 3, 4 and 5 repetitions not allowed?

Find the number of strings of 4 letters that can be formed with the letters of the word EXAMINATION?

There are 11 points in a plane. No three of these lies in the same straight line except 4 points, which are collinear. Find,
(i) the number of straight lines that can be obtained from the pairs of these points?
(ii) the number of triangles that can be formed for which the points are their vertices?

A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of
(i) exactly 3 women?
(ii) at least 3 women?
(iii) at most 3 women?

7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife has also 7 relatives 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen, so that there are 3 of men's relative and 3 of the wives relatives?

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw?

Prove that \(\frac { (2n)! }{ n! } \) = 2ⁿ (1.3.5...(2n - 1)).

How many different strings can be formed together using the letters of the word "EQUATION" so that
(i) the vowels always come together?
(ii) the vowels never come together?

By the principle of mathematical induction, prove that, for n\(\in \)N, cos α + cos(α + β) + cos(α + 2β)+...+ cos(α +(n - 1)β) = \(\left( \alpha +\frac { (n-1)\beta }{ 2 } \right) \times \frac { sin\left( \frac { n\beta }{ 2 } \right) }{ sin\left( \frac { \beta }{ 2 } \right) } \).