Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
The integers from 1 to 1000 are written in order around a circle. Starting at 1, every fifteenth numbers is marked (i.e., 1, 16, 31 etc.). This process is continued until a number is reached which has already been marked, then unmarked numbers are:
Step-by-Step Solution
Key Concept: The number of marked positions equals $\frac{N}{\gcd(15, N)}$ where $N = 1000$, since $\gcd(15, 1000) = 5$.
We mark every 15th number starting from 1 in a circle of 1000 numbers. The marked positions form an arithmetic sequence: $1, 16, 31, 46, ...$ which can be written as $1 + 15k$ (mod 1000). The sequence repeats when $1 + 15k \equiv 1 \pmod{1000}$, meaning $15k \equiv 0 \pmod{1000}$. We need $k = \frac{1000}{\gcd(15, 1000)} = \frac{1000}{5} = 200$. So exactly 200 numbers are marked, leaving $1000 - 200 = 800$ unmarked numbers.
Correct Answer: 1,2,3,4