Four children are running a race.
(i) In how many ways can the first two places be filled?
(ii) In how many different ways could they finish the race?

Count the number of three-digit numbers which can be formed from the digits 2, 4, 6, 8 if
(i) repetitions of digits is allowed.
(ii) repetitions of digits is not allowed

How many three-digit numbers are there with 3 in the unit place?
(i) with repetition
(ii) without repetition.

If ¹⁰P_r-1 = 2 \(\times\) ⁶P_r, find r.

A test consists of 10 multiple choice questions. In how many ways can the test be answered if
(i) Each question has four choices?
(ii) The first four questions have three choices and the remaining have five choices?
(iii) Question number n has n + 1 choices?

A student appears in an objective test which contain 5 multiple choice questions. Each question has four choices out of which one correct answer.
(i) What is the maximum number of different answers can the students give?
(ii) How will the answer change if each question may have more than one correct answers?

How many strings can be formed from the letters of the word ARTICLE, so that vowels occupy the even Places?

Find the distinct permutations of the letters of the word MISSISSIPPI?

Prove that ³⁵C₅ + \(\sum _{ r=0 }^{ 4\quad (39-r) }{ C_{ 4 }=^{ 40 }{ C }_{ 5 } } \)

In how many ways 4 mathematics books, 3 physics books, 2 chemistry books and 1 biology book can be arranged on a shelf so that all books of the same subjects are together.