METU Department of Mathematics Math 405 Combinatorics Spring 2010

QUIZ 2

Problem 1. There are three sets, A, B, and C, each set contains 20 elements. Each of the intersections A∩B, A∩C, and B ∩C contains 10 elements. The union A ∩ B ∩ C contains 33 elements. How many elements are in A ∪ B ∪ C ?

Problem 2. How many “words” can be obtained by permutation of the letters of “ANKARA” so that two letters “A” will stand next to each other, but not all 3 letters “A”?

Problem 3. In how many ways 5 people may sit at a round table?

Problem 4. How many summands are in the right-hand side of the trinomial formula (x + y + z)^{15} = ∑ a_{i,j,k} x^{i}y^{j}z^{k}? Find a_{4,5,6}.

Problem 5. In how many ways one can write 15 = a + b + c, where a, b, c ≥ 3?

Problem 6. In how many ways one can put 10 books on 3 shelves, so that there will be at least two books on each shelf ?