JEE Main 2020 (Online) 5th September Morning ShiftQuestion 1
The number of words, with or without meaning, that can be formed by taking $4$ letters at a time from the letters of the word `SYLLABUS' such that two letters are distinct and two letters are alike, is ________.
Permutation and Combination
Answer$240$
Solution
Concept. Pick which letter is doubled, pick two other different letters, then arrange the four letters with one repeat. Formulas used. (1) ${}^nC_r=\frac{n!}{r!\,(n-r)!}$(2) Arrangements of $4$ letters with one letter twice: $\frac{4!}{2!}$Step 1 - Letters. SYLLABUS $=$ S, S, L, L, Y, A, B, U. Repeated letters: S and L. Distinct letters: S, L, Y, A, B, U ($6$). Step 2 - The alike pair. S S or L L: $2$ ways. Step 3 - Two distinct letters. From the other $5$ distinct letters (the other of S/L can be used once): ${}^5C_2=10$. Step 4 - Arrange, by (2). $\frac{4!}{2!}=12$. Step 5 - Multiply. $2\cdot10\cdot12=240$. Answer: 240