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two truths and a lie ideas - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: This is equivalent to finding the number of ways to partition 6 distinguishable objects into exactly 4 non-empty, indistinguishable subsets. This is given by the Stirling numbers of the second kind, $S(6, 4)$. From known values, $S(6, 4) = 65$. Since the cabinets are indistinguishable, we do not multiply by permutations.

📌 Solution: This is an arithmetic sequence where the first term $ a = 3 $, the common difference $ d = 4 $, and the last term $ l = 99 $. To find the number of terms $ n $, we use the formula for the $ n $-th term of an arithmetic sequence:

📌 Solution: The smallest 4-digit number is 1000, and the largest is 9999. To find how many of these are divisible by 11, we find the smallest and largest multiples of 11 in this range.

📌 Now, count the number of terms in the sequence from 91 to 909 inclusive:

📌 Question: The average of $ 2x + 4 $, $ 5x - 3 $, and $ 3x + 7 $ is 20. What is the value of $ x $?

📌 Question: What is the probability that a positive integer less than or equal to 60 is a divisor of 60?

📌 Solution: First, count the positive divisors of 60. The prime factorization of 60 is:

📌 There are 60 positive integers less than or equal to 60. The probability that a randomly chosen integer from 1 to 60 is a divisor of 60 is:

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