1. (10 points.) Suppose that $n$ is an even integer greater than 0. How many bitstrings of
length $n$ have more 1's than 0's? Your answer need not be in closed form.
2. (10 points.) The Martian language can be written in the familiar Roman alphabet, which
has 21 consonants (B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, Z) and 5 vowels
(A, E, I, O, U). All Martian nouns begin with a consonant, followed by one or more vowel-
consonant pairs. However, no consonant can appear more than once in a single noun, and no
vowel can appear more than once in a single noun. For example, VUX, DIYOK, and
ZOBALUT are possible Martian nouns, but X, VUV, DIYIK, and ZEZELAL are not. How
many Martian nouns are possible? Your answer must be a number. Briefly explain your
answer.
3. (10 points.) Recent expeditions to Mars have cast doubt on whether the Martian language
can be written with exactly 21 consonants and 5 vowels. Suppose that Martian requires $c$
consonants and $v$ vowels, where $c > 0$, $v > 0$, and $c > v$. Express the number of possible
Martian nouns in terms of $c$ and $v$. Your answer need not be in closed form.
