Probability, Counting & Combinatorics
Season 17 of American Idol ended earlier this year. At the end of the season finale, each of the three judges was asked to share his/her opinion about who should win the contest; Laine or Alejandro. Each of the three judges responded and audiences around the country basically heard “Laine,” “Laine,” “Laine.” If each judge answered either “Laine” or “Alejandro,” how many additional ways could the judges’ responses have been heard?
It’s probably a little strange to think that some students started school last week, some have another couple of weeks of summer vacation and still others are just continuing with their year-round school schedule! The date of the beginning of the school year has been a topic of debate throughout the last decade (and even before) in many areas across the United States. But the trend is definitely leaning towards starting school earlier rather than later. Basically, in the last 10 years, the percent of public schools going back to school before Labor Day has increased by 63%
The Walton and the Brady families plan to vacation together in an 8-bedroom vacation home. John and Olivia will bring six children — boys, Jason, Ben and Jim-Bob, and girls, Mary Ellen, Erin and Elizabeth. Mike and Carol also will bring six children — boys, Greg, Peter and Bobby, and girls, Marcia, Jan and Cindy. John and Olivia will sleep in one of the bedrooms, and Mike and Carol will stay in another bedroom. Each of the remaining six bedrooms each will be occupied by a pair of children, with six boys using three bedrooms and six girls using three bedrooms.
Thursday is the 4th of July... here are a few problems relating to Independence Day!! While walking around at their town’s Independence Day Festival, Latrease and Hannah decide to buy lunch. Latrease bought two hot dogs and a soda for a total of $7.50. Hannah bought a hot dog, a soda, and a bag of chips for $5.50. If a bag of chips costs $1.00, how much does one hot dog cost?
At Walter Whitman Middle School, to celebrate the last day of school, Mrs. Hanks brought in treats for her class. She has chocolate bars, lollipops and gumballs. Mrs. Hanks tells each student they can choose any combination of three treats from what she has. How many different combinations of three treats can one of her students chose?
Last week the national competition concluded, and Daniel Mai from Massachusetts earned the title of MATHCOUNTS National Champion. Let’s look at some of the problems he had to solve on the way to the top!
Two opposite vertices of a certain square are located at (1, 6) and (−3, 1). If the line y = mx divides this square into two regions of equal area, what is the value of m? Express your answer as a common fraction.
On Sunday, May 12th, 224 of the nation’s most talented middle-school math minds will be in Orlando, FL for the 2019 Raytheon MATHCOUNTS National Competition. The stakes are high, and the problems will be tough. Here are a few problems national competitors solved in 2018.
The Pythagorean Triple Threat is a coed softball team. League rules require the batting order to alternate boy-girl-boy-girl… (or girl-boy-girl-boy…). If there are five girls and five boys on the team, how many different batting orders are possible?
In just a few days, the 2019 MATHCOUNTS State Competitions will begin, so let’s look back at some of the 2018 State Sprint round problems and solve them in preparation.
Over the weekend, the world’s most famous groundhog, Punxsutawney Phil, enjoyed his favorite day of the year – Groundhog Day! According to historical data on Stormfax® Weather Almanac’s website (stormfax.com/ghogday), Phil has predicted a long winter 104 times, and an early spring 18 times from 1887 to 2018 (no data for nine of these years). That makes the odds of Phil seeing his shadow, thereby predicting six more weeks of winter for this year, 52 to 9. Based on this information, what is the percent probability of Phil seeing his shadow?