#1

   In how many ways can the letters in the word computer be arranged so that there are no adjacent vowels?

#2

   Lee returns from the laundromat with a big bag of socks. There are 7 identical red socks, 9 identical white socks, 6 identical green socks, and 12 identical blue socks. If   Lee reaches in the bag and grabs socks without knowing the colors, how many must she grab to be sure of selecting a matched pair?

#3

    What is the sum of the coefficients in the expansion of

    (s + t + u + v + w)^15  ?

#4

   A domino shows a pair of numbers, each from 0 to 12. How many different dominoes could there be if we consider the pairs of numbers (b,a) and (a,b) to be the same domino?

#5

   What is the coefficient of  x^9y^11z^6  in  (x ­ y + z)^26 ?

#6

   Twelve identical dice are rolled once. How many possible outcomes are there?

#7

   In each of 10 identical bags are 6 marbles, one each of red, white, blue, green, pink, and yellow. Marbles of the same color are not distinguishable. In how many ways can we create a collection consisting of one marble from each bag?

#8

   Five Minutemen and eight Patriots are to be seated at a round table. If no two Minutemen may be seated together, how many ways can the group be seated?

#9

   Determine the number of integers from 1 to 2000 inclusive that are not divisible by any of 6, 10, and 35.

#10

   Mary  has budgeted $100 to give to local charities this year, spent in multiples of $5.  If  Mary  gives at least $5 to each of 10 local charities, how many ways can her donations be made? What if  Mary gives at least $5 to each of at most 10 local charities?

#11

   In how many ways can  8 Qs and  4 Ms be arranged so that no two Ms are adjacent?

#12

   How many non-negative integer solutions exist for the equation z + y + x + a + b + c = 28? Provide an example of a solution to this equation that is NOT a solution to the same equation over only the positive integers.

#13

   Suppose you have the nonsense word abbcccddddeeeee, where identical letters are indistinguishable. How many distinct arrangements are there for the letters of this word? How many ways can the letters in the word be arranged so that no two vowels are adjacent?

#14

   In a certain game 5 dice are rolled together. Two examples are (1,2,1,2,1) and (1,1,1,2,2). Each of these has the same outcome: three 1s and two 2s. How many different outcomes are possible in this game?

#15

   In how many ways can three balls of the same color be selected from an urn containing 5 red balls and 6 green balls?

#16

   In how many ways can you get three 6s, two 4s, four 1s, and one 2 in the roll of 10 distinct dice?

#17

   Determine the sum of the coefficients in the expansion of (x + y)^23.

#18

   How many uncollected terms are there in
(t₁ + t₂ + t₃ + t₄ )^20 ?

#19

   What are the collected terms in the expansion of (a + b + c)^10  that  have coefficients 10!/(3!5!2!)?

#20

   What is the absolute difference of the sum of the elements in rows 14 and 15 of Pascal's Triangle?

#21

   Use Pascal's Formula to express  ₄C₃  as the sum of as many non-zero terms (each expressed  in the form   _aC_b   as possible.

#22

   Seth had his choice from among 5 baseball cards labeled Mantle, Marris, Mays, Mitchell, and Morris.   If   Seth could choose from 0 to 5 cards, how many ways could he make his selection?

#23

   In how many ways can two integers be selected from the set {1,2,...,41} such that the sum of the two integers in an even number?

#24

   Show at least two combinatorial ways to describe the number 2^35.

#25

   Uncle David passed away and bequeathed 30 antique pliers to his nephew and niece, Ralph and Sophie. Sophie, a combinatorics prof at Redbird U, agreed to give Ralph all 30 pliers if he could answer this question:

If we count all the subsets of the 30 pliers that have an even number of pliers, and count all the subsets that have an odd number of pliers, which group will have more subsets?

Ralph really wants the pliers. How should he respond?

#26

   In how many ways can the 26 letters of the alphabet be arranged so that there are always exactly six letters between the letters c and d?

#27

   Three committees are to be chosen from 13 eligible members. If the committees must have 5, 3, and 2 people, respectively, and no person may serve on more than one committee, how many ways can the committees be formed?

#28

   What is the minimum number of human beings that must be assembled to insure that at least two of them share the same birth month?

#29

   A palindrome is a number that reads the same left to right as right to left, such as 32423. How many 7-digit palindromes are there? Assume that the number 0675760 is not a 7-digit number, because its left-most digit is 0.

#30

   What is the coefficient of x^14y^6 in the expansion of (x + 2y)^20 ?