How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5 ? if
(i) repetition of digits allowed
(ii) the repetition of digits is not allowed.

How many three-digit odd numbers can be formed using the digits 0, 1, 2, 3, 4, 5? if
The Repetition of digits is not allowed

How many three-digit odd numbers can be formed using the digits 0, 1, 2, 3, 4, 5? if 
The repetition of digits is allowed

Count the numbers between 999 and 10000 subject to the condition that there are
(i) no restriction.
(ii) no digit is repeated.
(iii) at least one of the digits is repeated.

To travel from a place A to place B, there are two different bus routes B₁, B₂, two different train routes T₁, T₂ and one air route A₁. From place B to place C there is one bus route say B'₁, two different train routes say T'₁, T'₂ and one air route A'₁. Find the number of routes of commuting from place A to place C via place B without using similar mode of transportation.

How many strings can be formed using the letters of the word LOTUS if the word
(i) either starts with L or ends with S?
(ii) neither starts with L nor ends with S?

By the principle of mathematical induction, prove that for n > 1
\(1^3 +2^3 +3^3 + .. +n^3=\left[n(n+1)\over 2\right]^2\)

Using the mathematical induction, show that for any natural number n > 2
\({1\over 1+2}+{1\over 1+2+3}+{1\over 1+2+3+4}+...+{1\over 1+2+3..+n}={n-1\over n+1}\)

Using the mathematical induction, show that for any natural number n
\({1\over 2.5}+{1\over 5.8}+{1\over 8.11}+...+{1\over (3n-1)(3n+2)}={n\over 6n+4}\)

8 women and 6 men are standing in a line.
(i) How many arrangements are possible if any individual can stand in any position?
(ii) In how many arrangements will all 6 men be standing next to one another?
(iii) In how many arrangements will no two men be standing next to one another?