The question about the word of "Mathematics". I have got a problem that contains three things;
i) How many different words can be formed by a rearrangement the letters of the word "mathematics"?
ii) How many of these words begin and end with t?
iii) How many words contain the character string "math" somewhere in that word?
Could you please tell me what to do?
(By the way, I've translated them into English from Danish. I know it causes misunderstanding.)
 A: No English word other than mathematics is an anagram of mathematics.
However, using the letters in the word mathematics, we can form the following words:

asthmatic, mismatch, attaches, thematic, schemata, matches, atheism, atheist, attache, aitches, chemist, satiate, miasma, attach, attics, asthma, tamest, thames, themas, theism, theist, tithes, ethics, itches, chaste, cheats, chimes, sachet, static, stitch, scathe, schema, maths, mates, matte, match, mamas, maims, maces, meats, mites, mitts, mimes, attic, aches, tames, tacit, taste, theta, teams, teats, teach, times, tithe, hates, haste, heats, heist, emits, ethic, imams, items, caste, chats, chasm, chase, cheat, chest, chime, chits, cites, smith, smite, state, steam, shame, math, mate, matt, mats, mama, maim, mace, mast, mash, meat, mesh, mite, mitt, mime, mica, mice, mist, ahem, aims, acme, acts, ache, aces, asia, tame, tact, that, thai, them, this, team, teat, teas, tech, test, time, ties, tits, tics, hams, hate, hats, hems, heat, hits, emit, eats, each, east, etch, imam, item, itch, ices, came, cams, cats, cast, cash, case, chat, chit, cite, cist, same, stem, stet, sham, semi, seam, seat, seth, sect, siam, site, scam, scat, mat, mac, mas, met, ate, aha, aim, act, ace, ash, tat, the, tea, tie, tit, tic, hmm, ham, hat, has, hem, him, hit, hic, his, ems, eat, eta, est, its, ice, cam, cat, sam, sat, sac, she, sea, set, sec, sit, sic, me, ms, am, at, ah, as, ha, he, hi, em, eh, im, it, is, a.


If with words you meant strings, as user1729 suggested, then, because the word mathematics contains $11$ characters, there are $11!$ strings.
However, because the letters m, a and t are repeated once, this changes the count to
$$
\frac{11!}{2!2!2!}=\frac{11!}{8}=4989600.
$$
To find the number of words that start and end with t, that is, words which fit the template
t _ _ _ _ _ _ _ _ _ t

we do as above and compute
$$
\frac{9!}{2!2!}=\frac{9!}{4}=90720.
$$
Finally, the number of strings which contain the sub-string math is $8!$ because, as John Adamski pointed out, we are only considering the characters in the string ematics, which contains $7$ unique characters, plus the string math as a whole.
