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Using inclusion-exclusion, the total is $ 9 + 7 - 2 imes 1 = 14 $.
Question: What is the smallest four-digit number divisible by 17?
The next integer is 59, so $ 17 imes 59 = 1003 $.
Solution: The sum of squares formula is $
Question: An ice cream truck offers 8 seasonal flavors and 5 artisanal toppings. If a customer selects 3 different flavors (flavor order doesnt matter) and 2 different toppings, how many unique combinations are possible?
Solution: First, compute the number of ways to choose 3 different flavors from 8. This is a combination:
Next, compute the number of ways to choose 2 different toppings from 5:
Since the choices of flavors and toppings are independent, the total number of combinations is:
Question: A chemical engineer is testing 6 catalysts in a series of reactions. If 4 catalysts are selected for a particular experiment (without replacement and order does not matter), what is the probability that catalyst A is included but catalyst B is not?
Question: A town is evaluating air quality sensors placed in 7 neighborhoods. If 3 sensors are randomly activated for calibration today, what is the probability that at least one sensor from the high-risk zone (which has 2 sensors) is selected?