METU Department of Mathematics Math 405 Combinatorics Spring 2010

QUIZ

Problem 1. In how many ways two squares can be selected from 8 × 8 chessboard so that they are not in the same row or column? (Give a formula, do not calculate.)

Problem 2. In how many ways 20 people can be divided into 10 pairs ? (Give a formula, do not calculate.)

Problem 3. In some kingdom there are 20 towns and each pair of towns are connected by one road. How many roads are there? (Give a formula and calculate the number.)

Problem 4. How many “words” can be formed from 3 different letters of a word “Istanbul” ? (Give a formula, do not calculate).

Problem 5. How many “words” can be obtained by permutations of letters of a word “essence” (all 7 letters should be used) ? (Give a formula, do not calculate).

Problem 6. Find the coefficient of x^{40}y^{60} in the expansion of (x + y)^{100} (do not calculate).

Problem 7. Find the coefficient of x^{4}y^{3}z^{2}t^{6} in the expansion of (x + y + z + t)^{15} (do not calculate).

Problem 8. How many numbers between 1 and 100 are not divisible neither by 3 nor by 5 ? (Calculate.)