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Understanding the Context
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📰 Solution: The sum of squares formula gives $S = rac{15 imes 16 imes 31}{6} = 1240$. Compute $1240 \mod 13$: $13 imes 95 = 1235$, so $1240 - 1235 = 5$. The remainder is $oxed{5}$.Question: A science educator is designing a lesson on probability using a digital simulation that generates random 6-digit numbers using only the digits 1, 2, and 3. What is the probability that a randomly generated number has exactly two consecutive digits that are the same?
📰 Solution: To solve this, we must count the number of 6-digit numbers using only the digits 1, 2, and 3 such that exactly one pair of consecutive digits are equal, and all other adjacent digits are different.
📰 Letâs denote the digits of the number as $ d_1, d_2, d_3, d_4, d_5, d_6 $, where each $ d_i \in \{1, 2, 3\} $.
