JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Permutations and Combinations

The word MANKIND has letters A, D, I, K, M, N, N (with N repeated twice). Arrange in alphabetical order: A, D, I, K, M, N, N. Count words before MANKIND: (1) Words starting with A, D, I, K: each contributes $\frac{6!}{2!}=360$ words, total $4\times 360=1440$. (2) Words starting with M, then A, D, I, K (before M-A-N): each contributes $5!=120$ words, but we need to account for the repeated N. Actually, words starting with MA but before MAN: none (next letter after A is N). Words starting with M, second letter before A: none. Words starting with MA, third letter before N: D, I, K give $3\times \frac{4!}{2!}=36$ words. (3) Words starting with MAN, fourth letter before K: D, I give $2\times 3!=12$ words. (4) Words starting with MANK, fifth letter before I: D gives $2!=2$ words. (5) Words starting with MANKI, sixth letter before N: none. (6) Words starting with MANKIN, seventh letter D: 1 word. Total = $1440+36+12+2+1+1=1492$.