JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Permutations and Combinations

The letters of UNIVERSE are U, N, I, V, E, R, S, E. The distinct vowels are E, I, U, with the letter E occurring twice, and the consonants are N, V, R, S. Since no letter may repeat, the two chosen vowels must be different letters, so they are selected from $\{E,I,U\}$ in $\left(\genfrac{}{}{0ex}{}{3}{2}\right)=3$ ways; the two identical copies of E cannot be used together. The two consonants are chosen from the four available in $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ ways. Each resulting set of four distinct letters can be arranged in $4!=24$ ways. The number of such four-letter words is $3\times 6\times 24=432$.