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Counting Techniques

Counting Techniques. Worst case. Ordered places or “for each”. 3. Permutation (Can be done by number 2). 4. Combination. Worst Case.

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Counting Techniques

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  1. Counting Techniques • Worst case. • Ordered places or “for each”. • 3. Permutation (Can be done by number 2). • 4. Combination.

  2. Worst Case Of the science books in a certain supply room, 50 are on botany, 65 are on zoology, 90 are on physics. 50 are on geology, and 110 are on chemistry. If science books are removed randomly from the supply room, how many must be removed to ensure that 80 of the books removed are on the same science?   (A) 81  (B) 159  (C) 166  (D) 285  (E) 324

  3. Ordered Places 1 Katie must place five stuffed animals--a duck, a goose, a panda, a turtle and a swan in a row in the display window of a toy store. How many different displays can she make if the duck and the goose must be either first or last? (A) 120 (B) 60 (C) 24 (D) 12 (E)  6

  4. Ordered Places 2 The president of a country and 4 other dignitaries are scheduled to sit in a row on the 5 chairs represented above. If the president must sit in the center chair, how many different seating arrangements are possible for the 5 people?  (A) 4  (B) 5  (C) 20  (D) 24  (E) 120

  5. Ordered Places 3 In how many arrangements can a teacher seat 3 girls and 3 boys in a row of 6 if the boys are to have the first, third, and fifth seats? (A)      6 (B)     9 (C)     12 (D)     36 (E)    720

  6. “FOR EACH” If a customer makes exactly 1 selection from each of the 5 categories listed below, what is the greatest number of different ice cream sundaes that a customer can create?  12 ice cream flavours 10 kinds of candy   8 liquid toppings   5 kinds of nuts   With or without whip cream. (A) 9600 (B) 4800 (C) 2400 (D) 800 (E) 400

  7. Ordered Places or Permutation Given a selected committee of 8, in how many ways, can the members of the committee divide the responsibilities of a president, vice president, and secretary? (A) 120 (B) 336 (C) 56 (D) 1500 (E) 100

  8. Ordered Places or Permutation How many four-digit numbers can you form using ten numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) if the numbers can be used only once? (A) 72 (B) 5000 (C) 4536 (D) 10 000 (E) 210

  9. Combination 1 From a group of 8 secretaries, select 3 persons for a promotion. How many distinct selections are there? (A) 56 (B) 336 (C) 512 (D) 9 (E) 200

  10. Combination 2 A person has the following bills: $1, $5, $10, $20, $50. How many unique sums can one form using any number of these bills only once? (A) 10 (B) 15 (C) 31 (D) 35 (E) 40

  11. Combination 3 and “For each” From a class consisting of 12 computer science majors, 10 mathematics majors, and 9 statistics majors, a committee of 4 computer science majors, 4 mathematics majors, and 3 statistics majors is to be formed. How many distinct committees are there? (A) 1000 (B) 5270 (C) 9 327 (D) 12 345 (E) 8 731 800

  12. Combination 4 and “For each” A three-person committee must be chosen from a group of 7 professors and 10 graduate students. If at least one of the people on the committee must be a professor, how many different groups of people could be chosen for the committee? (A)  70 (B)  560 (C) 630 (D) 1,260 (E) 1,980

  13. Combination 5 and “For each” There are 11 top managers that need to form a decision group. How many ways are there to form a group of 5 if the President and Vice President are not to serve on the same team? (A) 200 (B) 315 (C) 378 (D) 425 (E) 498

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