Permutations and Combinations
Permutations with Identical Objects
JEE Advanced 2017 Paper 1
Grade 11
Question:
Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let $x$ be the count with no letter repeated, and $y$ be the count where exactly one letter is repeated exactly twice and no other letter is repeated. Then $y/x$ equals _____.
Step-by-Step Solution
Key Concept: $x = 10!$ (all 10 distinct letters in 10 positions). For $y$: choose the repeated letter (10 ways), choose 8 of the remaining 9 letters ($\binom{9}{8} = 9$ ways), arrange all 10 positions with one letter repeated ($10!/2!$ ways). $y = 10 \times 9 \times 10!/2 = 45 \times 10!$. So $y/x = 45$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 45