# Sagetex: Combinatorics/Probability

Problem Type 1: An urn contains r red balls and b black balls. If k balls are chosen, what's the probability that t balls are red?

Problem Type 2: Enter a word in upper case, such as CALCULUS (shown above) and ask for the number of different arrangements of all the letters.

Problem Type 3: How many subsets are there of a given set? How many subsets have a particular size?

Problem Type 4: How many functions are there from an m element set to an n element set? How many of these functions are one-to-one?Download file: CombProb4 (.tex)

Problem Type 5: (Inclusion-Exclusion) How many numbers between 1 and N are divisible by primes $latex p_1,p_2,p_3$?

Problem Type 6: A committee will be made from 5 English teachers, 3 math teachers, and 4 history teachers. If the committee must have 2 teachers from different disciplines then how many committees are there?

Problem Type 7: How many nonnegative integers less than 100000
contain the digit 5?

Problem Type 8: A high school committee of 6 is to be made from 32 boys and 41 girls. Within this set of students there are 2 senior boys and 3 senior girls. How many committees of 3 boys and 3 girls are there that contain at least one senior boy and one senior girl?

Problem Type 9: Suppose the probability of having a girl is 1/3 and having a boy 2/3. Find the probability that a family of 5 has:

a) exactly 2 boys

b) at least 1 boy and 1 girl

Problem Type 10: Suppose a system has 6 independent components, each of which is equally likely to work. Now suppose the system works if and only  5 components work. What is the probability the system works.Download File: Comb10  (.tex)

Problem Type 11: Find the value of C(9,6) and P(9,3). Show your steps.

Problem Type 12: Find all the divisors of N.

Problem Type 12: Three darts are thrown at the dartboard. A score is given for each region where the dart lands and the total score is just the sum of the $3$ dart scores. Assume all three darts hit the dartboard. How many different total scores are there? Enumerate them with by drawing a tree diagram.