There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees
(i) a particular teacher is included?
(ii) a particular student is excluded?

If ⁽ⁿ⁺²⁾P₄ = 42 \(\times\) ⁿP₂, find n.

How many 'letter strings' together can be formed with the letters of the word "VOWELS" so that
(i) the strings begin with E
(ii) the strings begin with E and end with W.

There are 15 candidates for an examination. 7 candidates are appearing for mathematics examination while the remaining 8 are appearing for different subjects. In how many ways can  they be seated in a row so that no two mathematics candidates are together?

How many numbers can be formed using the digits 1, 2, 3, 4, 2, 1 such that, even digits occupies even place?

Prove that ¹⁰C₂ + 2 x ¹⁰C₃ + ¹⁰C₄ = ¹²C₄

In rating 20 brands of cars, a car magazine picks a first, second, third, fourth and fifth best brand and then 7 more as acceptable. In how many ways can it be done?

If a set of m parallel lines intersect another set of n parallel lines (not parallel to the lines in the first set), then find the number of parallelograms formed hi this lattice structure.

How many diagonals are there in a polygon with n sides?

Using the Mathematical induction, show that for any integer
n\(\ge\) 2, 3n² > (n + 1)²