Problem 1. How many a) two-letter; b) three-letter "words" are there? Use letters of the English alphabet. Send a solution

Problem 2. How many different necklaces made of a) three different colored beads; b) two red and two blue beads; c) three red and two blue beads are there? Send a solution

Problem 3. In how many ways can we choose two team leaders and one team captain out of ten people? Send a solution

Problem 4. In how many ways can we choose three team leaders out of a) five people; b) seven people; c) ten people? Send a solution

Problem 5. In how many ways can we seat five people on a bus if there are a) 4; b) 5; c) 6; d) 7 empty seats in the bus? Send a solution

Problem 6. Seven mathematicians came to an amusement park. They want to a) go on a train ride, which consists of seven single-seated wagons; b) on a seven-seated carousel; c) on a train ride, which consists of ten single-seated wagons; d) on a ten-seated carousel. In how many ways can they do that? Send a solution

Problem 7. In how many ways can you walk from the bottom left corner to the upper right corner of the square: a) $2\times 2$; b) $3\times 3$; c*) $5\times 5$. You are only allowed to move up or right along the sides of the square. Send a solution

Problem 8. In how many ways can number $5$, $10$, $20$ be written as a sum of a) two; b) three positive integers? Send a solution

Problem 9. In how many ways can you place parentheses in the following expression $a+b-c\cdot d$? Send a solution

Problem 10. a) Show that there are as many subsets in the set {a, b, c, d} as there are relations between the set {a, b, c, d} and the set {0, 1} that associates to every element of a first set exactly one element of the second set. b) Show that that number is equal to the number of sequences of zeros and ones of length five. Send a solution

Problem 11. How many different sets of beads that can make up exactly two different necklaces are there? Send a solution

Problem 12. In Springfield, bus tickets have four-digit numbers. The inhabitants believe that the tickets for which the sum of the first two digits equals the sum of the last two digits bring luck. How many tickets bring luck in Springfield? Send a solution

Problem 13. In how many ways can a Ferris wheel a) consisting of $7$ gondolas be painted in $3$ colors; b) consisting of $10$ gondolas be painted in $2$ colors? You don't have to use all the colors for painting. Send a solution

Problem 14. You cut a regular a) hexagon; b) heptagon; c) octagon into triangles with non-crossing diagonals. How many different sets of triangles can you obtain? Send a solution

Problem 15. How many different dice are there? Each face is numbered 1 through 6. Send a solution