# 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)

Nalin Pithwa

## 3 thoughts on “Two introductory ptoblems in combinatorics”

1. rahul sijwali

636

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• please explain as much as possible, though

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2. rahul sijwali

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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