MATHCOUNTS Class Counting Probability Reagan Choi Part 1

MATHCOUNTS Class Counting & Probability Reagan Choi

Part 1: Basic Techniques

Basic Counting Jake has six red shirts and four pairs of jeans. How many ways can he choose an outfit to wear? Jake gets four blue shirts in addition to what he had before. Now, how many ways can Jake choose an outfit to wear?

Permutations and Combinations Ten people are in a club. In how many ways can they stand in a line? Ten people are in a club. In how many ways can we choose one president, one vice-president, and one secretary, if nobody is allowed to have more than one role? Ten people are in a club. In how many ways can we choose exactly three people to go on a committee?

Permutations and Combinations, cont. Factorial: 1*2*3*. . . *n = n! Permutation: n. Pk = the number of ways to select k items from n total, with the k items ordered. Formula: n. Pk = n*(n-1)*(n-2)*. . . *(n-k+1) = (n!)/(n-k)! Combination: n. Ck = the number of ways to select k items from n total, where order doesn’t matter. Formula: n. Ck = (n!)/(k! * (n-k)!) =

Casework How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

Complementary Counting How many positive 4 -digit numbers contain the digit 6 or the digit 7?

Exercises How many ways are there to select five distinct one-digit positive integers such that at least two of the integers have a difference of 1?

Exercises One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has {1, 2, 3, 6, 9} as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?

Part 2: Techniques

Example There are 300 people at my high school that attend some random high school, and everyone is in either Math Club, Drama Club, or Book Club. If there are 156 people in Math Club, 126 people in Drama Club, 106 people in Book Club, 44 people in both Math Club and Drama Club, 31 people in both Math Club and Book Club, and 33 people in both Drama Club and Book Club. How many people attend all three clubs?

Stars and Bars How many ways are there to give ten candies to five children? C|CC||CCCC

Example How many ordered quintuples of positive integers, (a, b, c, d, e), satisfy a*b*c*d*e=4096?

Exercises How many ways can we select 6 distinct positive integers between 1 and 20, inclusive, so that no two of these integers have a difference of 1? How many ordered pairs of integers (a, b, c) are there such that a, b, c are divisors of 432, and a*b*c is also a divisor of 432?

Homework 1. A combination lock uses five integers in the combination, and the dial is numbered with the integers 0, 1, 2, and 3. If consecutive numbers in the combination cannot be the same, then how many possible combinations are there? 1. In how many ways can Alice, Bob, Charles, David, and Eva stand in a line, if Bob and Charles insist that they stand next to each other?

Homework 3. If a 1, a 2, a 3, a 4, a 5, are positive integers that sum to 25, and ai is at least i for i=1, 2, 3, 4, and 5, then how many ways can we choose such ai? 4. In a school of 100 students, 90 study English, 75 study Spanish, and 42 study French. Every student must study at least one of the three languages. What is the least possible number of students who could be studying all three languages?

Homework 5. How many ways are there to rearrange five red, five blue, and five green balls in a row so that no two green balls lie next to each other? 6. You are given eight distinguishable rings. How many ways are there to choose five of these rings to put on your fingers (excluding the thumb), if fingers are allowed to have no rings, and the order of rings on a finger matters?

Homework 7. A positive integer abcd (a, b, c, d are digits) is called “snakelike” if the inequalities a<b, b>c, c<d are true. How many snakelike positive integers between 1000 and 9999 have four distinct digits?