Practice Problems
Q-1) There are p intermediate stations on a railway line between 2 points A and B. In how many ways can a train stop at 3 of the stations such that no 2 of the stopping stations are consecutive?
Q-2) How many different numbers smaller than \(2\times 10^8\) can be formed by the digits 0,1,2 such that they’re divisible by 3?
Q-3) Every man who has lived on earth has made a certain number of handshakes. Show that the number of men who have made an odd number of handshakes are even.
Q-4) Find the number of ways in which the candidates \(A_1,A_2,A_3,..,A_{10}\) can be ranked such that \(A_1\) is always ranked higher than \(A_2\).
Q-5) Find the number of ways to choose 3 numbers from 1,2,3..,n such that they are in A.P. Consider separate cases for odd and even values of n.
Q-6) If 3 unbiased dices are thrown together, find the number of ways in which the sum of the numbers appearing is \(n\), with \(9 \le n\le 14\).
Q-7) Let \(a\le b\le c\) be integers denoting the sides of a triangle. If \(c\) is given, find the number of different triangles that are possible. Consider separate cases for odd and even values of \(c\).
Q-8) From a set of \(3n\) consecutive positive integers, find the number of ways of choosing 3 numbers such that their sum is divisible by 3.
Q-9) Let \(n,k\) be natural numbers with \(n \ge \frac{k(k+1)}{2}\). Given that \(x_i\ge i\), find the number of integer solutions of \(\sum\limits_{i=1}^{k}x_i=n\).
Q-10) Find the tenth and hundredth digit of 1!+2!+3!+…97!