JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Permutations and Combinations

In LETTER the consonants are L, T, T, R and the vowels are E, E. All six letters arrange in $\frac{6!}{2!2!}=180$ ways, allowing for the repeated T's and E's. Treating the two E's as one block makes five units, the block together with L, T, T, R, which arrange in $\frac{5!}{2!}=60$ ways and count the cases where the vowels are together. Subtracting, the number of words with the vowels never together is $180-60=120$.