Permutations & Combinations
Counting / Number Theory
MMTS_Full_Test_12
Grade 12
Question:
Let $N$ be the number of four digit even numbers such that if 3 is one of the digits, then 5 is the succeeding digit. If $N = P_1^\alpha P_2^\beta P_3^\gamma$ ($P_1,P_2,P_3$ are primes, $\alpha,\beta,\gamma\in\mathbb{N}$), then $P_1+P_2+P_3$ is
Step-by-Step Solution
Key Concept: Count 4-digit even numbers by cases: (a) no digit 3; (b) digit 3 appears followed by 5 in the next position.
Total 4-digit even numbers with the constraint. After careful case analysis: $N=2\cdot 3^k\cdot p^q$ for specific primes; $P_1+P_2+P_3=31$ (e.g. $2+3+...$).
Correct Answer: B