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Permutation and Combination
11. From a group of 10 professors and 6 assistant professors, a management institute desires to send a delegation of 8 persons consisting of 5 professors and 3 assistant professors to the IIMs annual meet. If Prof. Balamurali refuses to be in the delegation if Assistant Prof. Sheshadri is included in the delegation, then in how many ways can the delegation be formed?
12. Sixteen villages in a district are divided into 4 zones with four villages per zone. The telephone department intends to connect villages such that every two villages in the same zone are connected with two direct lines and every two villages belonging to different zones are connected with one direct line. How many direct telephone lines are required?
13. A certain number of students participated in a chess tournament. In 66 games both players were girls and in 240 games one was a girl and the other a boy. The number of games in which both players were boys is:
14. If all possible 5-digit numbers using digits 1,2,3,4,5 without repetition are arranged in ascending order, what is the position of 21354?
15. In an ‘akhada’, m wrestlers stand around a circle. Each possible pair of persons not standing next to each other plays a match for 1 minute 30 seconds. If the total time taken is 30 minutes, then m equals:
16. Let k be an integer such that the sum of digits of k is 2 and 10⁹ < k < 10¹⁰. How many values can k have?
17. Nine points lie on a straight line and 10 points lie on another parallel line. How many triangles can be formed?
18. Ten points are plotted in a plane such that no three are collinear. Four are joined to each of the remaining six, and each of the six is joined to exactly five points. How many line segments are formed?
19. In the figure, cars can travel only eastwards or southwards. In how many ways can a car travel from P to Q?
20. Using the first ten letters of the English alphabet, how many strings of 4 letters can be formed such that each H is followed by A? (Repetition allowed)
