Permutations & Combinations
Counting numbers with divisibility conditions using inclusion-exclusion
nta_pyq_2023_jan
Grade None
Question:
The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is
Step-by-Step Solution
Key Concept: Use inclusion-exclusion: |div by 3 or 4| = |div by 3| + |div by 4| − |div by 12|, then subtract those divisible by 48.
3-digit numbers: 900 total. Div by 3 = 300, div by 4 = 225, div by 12 = 75. Div by 3 or 4 = 300+225−75 = 450. Multiples of 48 in [100,999]: 144,192,...,960 $\to$ 18 terms. Answer = 450−18 = 432.
Correct Answer: 2