Two introductory ptoblems in combinatorics

Due: Prof. Titu Andreescu’s 104 Combinatorial Problems from Training of USA IMO Team.

Problem 1:

Call a 7-digit telephone number d_{1}d_{2}d_{3}-d_{4}d_{5}d_{6}d_{7} memorable if the prefix sequence d_{1}d_{2}d_{3} is exactly the same as either of the sequences d_{4}d_{5}d_{6} or d_{5}d_{6}d_{7} (possibly both). Assuming that each d_{i} can be any of  the ten decimal digits 0, 1, 2, 3, \ldots, 9, find the number of different memorable telephone numbers. (AHSME 1998)

Problem 2:

Two of the squares of a 7 \times 7 checkerboard are painted yellow and the rest are painted green. Two colour schemes are called equivalent if one can be obtained from the other by a rotation in the plane of the board. How many inequivalent colour schemes are possible? (AIME 1996)

Kindly share your solutions, comments,

Nalin Pithwa





3 thoughts on “Two introductory ptoblems in combinatorics

  1. question_2 we give yellow color any two blocks of the board there are 49*48 ways of doing it and the rest be green then observe that due to the square figure of the board there are 4 Rotationally symmetric case of every single 49*48 cases therefore (49*48)/4 therefore 588 cases


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