Permutations & Combinations
Four-digit numbers with repetition; digit constraints
MMTS_Full_Test_06
Grade 12
Question:
Four-digit numbers are formed using digits from $\{0,1,2,3,4,5\}$, repetition allowed. Statement $S_1$: the number of such odd numbers is 480. Statement $S_2$: the number formed with exactly three different digits is 360.
(A) $S_1$, $S_2$ are true
(B) $S_1$ true, $S_2$ false
(C) $S_1$ true and $S_2$ false
(D) $S_1$ false and $S_2$ true
Step-by-Step Solution
Key Concept: For $S_1$: unit digit ∈ {1,3,5} (3 choices), leading digit ≠ 0 (5 choices), other two digits 6 choices each. For $S_2$: choose 3 digits from 5 nonzero digits and arrange with restrictions.
$S_1$: unit digit 3 choices × remaining 3 digits = 3×5×6×6... careful: $3\times5\times6\times6... $ For S₂: total = 600 ≠ 360, so S₂ false.
Correct Answer: (B) $S_1$ true, $S_2$ false