1. In how many different orders can five runners finish a race if no ties are allowed?

2. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

3. In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

4. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?

5. A bag contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the bag, if at least one black ball is to be included in the draw?

6. How many bit strings of length 10 contains: a) Exactly four 1s? b) At most four 1s? c) At least four 1s? d) An equal number of 0s and 1s?

7. How many ways are there for four men and five women to stand in aline so that a) All men stand together? b) All women stand together?

8. suppose there are nine students in a discrete math class in a small college. Show that the class must have at least five male students or at least five female students

9. Show that In a packed Carnegie Hall performance, there will be two people who have the same first and last initials. (The main hall at Carnegie seats roughly 2,800)

10. Imagine a certain college has 6,000 American students, at least one from each of the 50 states. Show there must be a group of 120 students coming from same state. Please show your steps and submit your scanned / typed solution thru eLearning.