Math 106 – Exam #1 - Review Problems 1. (a) (b) (c) (d) (e) Evaluate each of the expressions displayed, and write each answer as a single numerical value. P(10,0) P(9,4) P(7,7) C(7,4) C(15,0) Expand (3x – 2) 7 with like terms combined together. 2187x 7 – 10206x x 5 – 22680x x 3 – 6048x x – Find the coefficient of x 4 y 12 in the expansion of (4x 2 + 5y 3 ) ,000 x 4 y 12 C(6,4) (4x 2 ) 2 (5y 3 ) 4 = 4. (a) Five boys and ten girls are walking to a playground. Write a ready-to-calculate formula for the number of ways each task listed can be done, but you do not have to actually calculate the final answer. The children will line up in single file to enter the narrow gateway to the playground. 15!

(b) (c) (d) (e) (f) (g) (h) (i) (j) The children will line up in single file to enter the narrow gateway to the playground with children of the same sex staying together. The children will line up in single file to enter the narrow gateway to the playground with all the girls staying together. 2!  5!  10! 6!  10! The children will line up in single file to enter the narrow gateway to the playground, but Judy and Jane must walk next to each other. 2!  14! The children will line up in single file to enter the narrow gateway to the playground, but Judy and Jane refuse to walk next to each other. 15! – 14!  2! At the playground, the children will sit in a single row of 20 swings. At the playground, the children will sit on a circular merry-go-round with 15 seats. At the playground, the children will sit on a circular merry-go-round with 15 seats, but Judy and Jane must sit next to each other. C(20,15)  15! 14! 13!  2! At the playground, the children will sit on a circular merry-go-round with 15 seats, but Judy and Jane refuse to sit next to each other. 14! – 13!  2! At the playground, the children will sit on a circular merry-go-round with 15 seats with children of the same sex staying together. 9!  10  5! or 4!  5  10!

5. (a) (b) (c) (d) (e) (f) (g) The Computer Science Club consists of eight seniors, seven juniors, five sophomores and six freshmen. Write a ready-to-calculate formula for the number of ways each task listed can be done, but you do not have to actually calculate the final answer. The four offices of President, Vice President, Secretary, and Treasurer are to be filled, but no person can hold more than one office. P(26,4) A committee of four must be chosen. C(26,4) A committee of four must be chosen containing exactly two seniors. C(8,2)  C(18,2) A committee of four must be chosen containing at most two seniors. C(8,0)  C(18,4) + C(8,1)  C(18,3) + C(8,2)  C(18,2) A committee of four must be chosen containing at least one senior. C(8,1)  C(18,3) + C(8,2)  C(18,2) + C(8,3)  C(18,1) + C(8,4)  C(18,0) or C(26,4) – C(18,4) A committee of four must be chosen so that each member is from the same class. A committee of eight must be chosen so that there are exactly two members from each class. C(8,4) + C(7,4) + C(5,4) + C(6,4) C(8,2)  C(7,2)  C(5,2)  C(6,2) 6.A child has cubic blocks with letters on them. The child is playing with four blocks with an “A” on them, five blocks with an “E” on them, four blocks with an “I” on them, four blocks with an “O” on them, and three blocks with a “U” on them. If the child is going to place these blocks in a row, how many arrangements are possible?

20! —————— = 244,432,188,000 4! 5! 4! 4! 3! 7. (a) (b) (c) (d) Each of six dice is a different color, and each die is a fair, standard, six-sided die. The five dice are to be rolled simultaneously. Calculate the percentage of times we would expect to see no number repeated. Calculate the percentage of times we would expect to see at least one number repeated. 6! — = = 1.54% 6 6 6! 1 – — = = 98.46% 6 6 Calculate the percentage of times we would expect to see no fives. Calculate the percentage of times we would expect to see at least one five. 5 6 — = = 33.49% – — = = 66.51% 6 6