Name:      

Directions: Show all work. Unless otherwise directed, you may leave your answer in terms of factorials, falling factorials, and binomial/multinomial coefficients. Any sums or products should be as simple as possible.

1. [2.5 points] How many $3$-element subsets of $\{a,b,c,d,e,f,g,h\}$ are there? Give an explicit numerical answer.
2. [2.5 points] A group of $50$ people must be split into two teams of size $5$ and five teams of size $8$. How many ways can this be done?
3. [2.5 points] A random lattice path of length $2n$ is generated by starting from $(0,0)$. At each step, we increase the $x$-component by $1$ with probability $1∕2$ and we increase the $y$-component by $1$ with probability $1∕2$. What is the probability the lattice path ends at $(n,n)$?
4. [2.5 points] A standard deck of cards has one card for each of the suit/rank pairs. The suits are spades, hearts, diamonds, and clubs; the ranks are ace, $2$ through $10$, jack, queen, and king. What is the probability that a poker hand (i.e. a set of $5$ cards) has at most $2$ cards in each suit?