Problem 1
How many arrangements of the letters in the word MISSISSIPPI do not have consecutive I's?
Problem 2
12 identical baseballs are distributed to 4 people, A, B, C, and D. If each distribution is equally likely, what is the probability that person A gets at least 5 balls?
Problem 3
How many words of 6 letters either begin aa or end zz?
Problem 4
How many integers less than 1000 are not divisible by 2,3,5 and 7? If $n$ has the prime factorization $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ how many positive integers $\lt n$ are relatively prime to $n$?
Problem 5
Explain, using 2-subsets of an n-set, why $$ \binom{\binom{n}{2}}{2}= \binom{n}{4}*\binom{4}{2}/2+3*\binom{n}{3}. $$
Problem 6
How many 5 digit numbers are not palindromes?
Problem 7
A box has 12 blue balls and 17 red balls. If 5 balls are drawn at random from the box, what is the probabillity that 2 blue and 3 red balls have been chosen?
Problem 8
Using the binomial theorem find $$ \sum_{k=0}^n \binom{n}{k}\binom{n}{n-k}*(-1)^{n-k}. $$
Problem 9
26 boys and 26 girls, with last names beginning with letters A to Z, are in 2 lines facing one another. What is the probability that all boys never directly face a girl who has the same last initial letter?
Problem 10
What is the probability that a random 5-element subset of {1,2,3,...,17} does not contain consecutive integers?
Problem 11 Let $a_n=S(n,2).$ Use the Stirling number recurrence to find a recurrence for $a_n$, solve that recurrence to find $S(n,2).$
Problem 12 Suppose that $$ a_n=a_{n-1}+5a_{n-2} $$ for $n \ge 2$ and $a_0=1, a_1=1.$ Find the generating function for $a_n$, and use it to give an explicit formula for $a_n$.
Problem 13 Find the generating for $a_n$,
(a) the number of ways n identical balls may be placed into 5 distinct boxes, each box non-empty,
(b) the number of ways n identical balls may be placed into 4 distinct boxes, box #1 containing between 4 and 8 balls, box #2 an even number of balls, box #3 a number of balls which is 1 modulo 3, box #4 and box #5 have no restriction.
Problem 14 Find all non-negative integers $n$ such that $F_{n+1}\ge 2F_n.$
Problem 15 Suppose the first difference of a sequence $a_n$ is a fixed sequence, $$ a_n-a_{n-1}=b_n, n \ge 1, $$ Show that the solution is $$ a_n=b_n+b_{n-1}+\cdots +b_1+a_0. $$ Conclude that if $b_n$ is a polynomial of degree $k$ in $n$, then $a_n$ is a polynomial of degree $k+1$ in $n$.