JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Permutations and Combinations

The letters are $P$ (three times), $Q$ (twice), $R$ (twice) and $S,T,U,V$ once each, seven distinct letters in all; the four-letter arrangements are grouped by repetition pattern. Three alike with one different: only $P$ can repeat thrice, the odd letter is chosen in $6$ ways, and $\frac{4!}{3!}=4$ arrangements each give $6\cdot 4=24$. Two pairs: the doubled letters are chosen from $\{P,Q,R\}$ in $\left(\genfrac{}{}{0ex}{}{3}{2}\right)=3$ ways with $\frac{4!}{2!2!}=6$ arrangements, giving $18$. One pair with two distinct singles: the pair is one of $\{P,Q,R\}$ ($3$ ways), the two singles are chosen from the remaining six in $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ ways with $\frac{4!}{2!}=12$ arrangements, giving $3\cdot 15\cdot 12=540$. All four distinct: $\left(\genfrac{}{}{0ex}{}{7}{4}\right)\cdot 4!=35\cdot 24=840$. The total is $24+18+540+840=1422.$