IMC training 2016 (2)

Methods of Counting

Problem 1. How many subsets are there in a set of size 10?

Problem 2. Find the number of squares contained in an 10\times 10 squares array.

Problem 3. A team is to be chosen from 4 girls and 6 boys. The only requirement is that it must contain at least 2 girls. Find the number of different teams that may be chosen.

Problem 4. You wish to give your Aunt Mollie a basket of fruit. In your refrigerator you have six oranges and nine apples. The only requirement is that there must be at least one piece of fruit in the basket (that is, an empty basket of fruit is not allowed). How many different baskets of fruit are possible?

Problem 5. Find the number of ways 30 identical pencils can be distributed among three girls so that each gets at least 1 pencil.

Problem 6. There 7 boys and 3 girls in a gathering. In how many ways can they be arranged in a row so that

1) the 3 girls form a single block?

2) the two end-possitions are occupied by boys and no girls are adjacent?

Problem 7. Between 20000 and 70000, find the number of even integers in which no digit is repeated.

Problem 8. A 2\times 3 rectangle is divided into six unit squares A,B,C,D,E and F. Each of these unit squares is to be colored in one of 6 colors such that no two adjacent squares have the same colors. Find the number of different rectangles.

Problem 9. An equilateral triangle ABC of side 10 is divided in to 100 equilateral triangles of side length 1 by lines parallel to its sides.

1) How many equilateral triangles are there?

2) How many rhombuses are there?

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