In how many different ways can the letters of the word '$CYCLE$' be arranged?

$A$ student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select at least one book is $63$,then the value of $n$ is

How many numbers can be formed from the digits $1, 2, 3, 4$ when the repetition is not allowed?

Six '$X$'s have to be placed in the squares of the figure such that each row contains at least one '$X$'. In how many different ways can this be done?

Find the number of ways in which two Americans,two British,one Chinese,one Dutch,and one Egyptian can sit at a round table so that persons of the same nationality are separated.

How many $3$-digit odd numbers can be formed from the digits $1, 2, 3, 4, 5, 6$ when
$(i)$ repetition of digits is not allowed
(ii) repetition of digits is allowed?