Math 281 - 01 (Test 1 SAMPLE) 1. The letters A, B, C, D and E are used to form string (or word) of length 4. (a) How many strings can be formed if we allow repetitions? (b) How many strings can be formed if we do not allow repetitions? (c) How many of the strings contains the sub-string DC (in any order) if repetition is not allowed? 2. Two dice are rolled, one blue and one red. How many outcomes have either the blue die 3 or an even sum (i.e., the sum of the numbers from the blue and the red dice is even)? 3. Let A and B be two sets of cardinalities 5 and 7, respectively. (i) How many functions are there from A into B? (ii) How many one-to-one functions are there from A to B? 4. (a) In how many ways can the letter in the word TEETOTALLER be arranged in a row? (b) In how many ways can eight distinct books be divided among 3 students if Bill gets 4 books and Shizo and Martin each gets 2 books? (c) A shipment of 100 compact discs contain seven defective discs. In how many ways can we select five compact discs that contain more defective discs than non-defective discs? (d) How many ways are there shelve 7 distinct Mathematics books and 4 distinct Computer Science books in a row so that no two Computer Science book stand next to each other? (e) In how many ways can three married couples be seated around a circular table? 5. (a) How many non-negative integer solutions does the equation x 1 x 2 x 3 x 4 11 have? (b) In how many ways can 12 identical doughnuts be distributed among four children, if each child gets at least one doughnut? 6. How many positive integers between 100 and 999 inclusive (a) are there? (b) are divisible by 3? (c) are not divisible by 5? (d) are divisible by 3 or 5? 7. How many permutations of the letters in MISSISSIPPI have no consecutive S’s? 8. (a) How many ways can a committee of five be chosen from 10 eligible individuals? (b) How is the result in (a) changes if two of the individuals desire to serve together on the committee or not at all? 1
9. A Social Security number is a sequence of nine digits, for example, 080-55-1617. (a) How many possible Social Security numbers are there? (b) How many possible Social Security numbers are there satisfying the following restrictions? (i) The digit 0 is not used. (ii) The sequence neither begins nor ends with the digit 0. (iii) No digit is repeated. 10. (a) How many five-letter strings can be formed using the letters a, b, c, d (with repeated letter allowed)? (b) How many of the strings in part (a) do not contain the substring “bad”? 11. Determine the coefficients of (a) x 3 y 5 in 2x y 8 (b) xy 3 z 6 in 2x y z 10 12. A committee of 12 is to be selected from 10 men and 10 women. In how many ways can the selection be carried out if (a) there is no restriction? (b) there must be equal number of men and women? (c) there must be an even number of women? (d) there must be more women than men? (e) there must be at least eight men? 13. In how many ways can 11 distinct books be divided among four students, A, B, C and D, if students A, B and C get two books each and student D gets five books? 14. Show that every set of 15 socks chosen among 14 pairs of socks contain at least one matched pair? 15. (a) How many non-negative integer solutions does the equation x 1 x 2 x 3 x 4 x 5 9 have? (b) How many integer solutions of the equation x 1 x 2 x 3 x 4 x 5 9 satisfy x 1 1, x 2 3, x 3 0, x 4 2, x 5 0? 16. Professor Euclid is paid every other week on Friday. Show that in some month she is paid three times.
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