Permutations and Combinations
Permutations with Identical Objects
JEE Main 2019
Grade 11
Question:
The number of 4-letter words (with or without meaning) that can be formed from the letters of the word MATHEMATICS is:
(1) $2454$
(2) $2466$
(3) $2500$
(4) $2640$
Step-by-Step Solution
Key Concept: MATHEMATICS has 8 distinct letters with M, A, T each appearing twice. Case 1 (all distinct): $^8C_4 \times 4! = 1680$. Case 2 (one pair + 2 distinct): $^3C_1 \times ^7C_2 \times \frac{4!}{2!} = 756$. Case 3 (two pairs): $^3C_2 \times \frac{4!}{2! 2!} = 18$. Total $= 1680 + 756 + 18 = 2454$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)