Find the number of 2 lettered words, with or without meaning, that can be formed out of the letters of the word TIME, where the repetition of the letters (a) is not allowed (b) is allowed.

Hint: To solve this question, we will assume a word with only two letters and then we will find how many choices each letter has in that word.

Complete step-by-step answer:
In this question, we have to form a word with two letters coming from the letters of the word TIME. To solve this question, we will check how many choices of letters each letter has in 2 lettered words. In the question given, we are given two conditions about the repetition of words. So we will have each condition separately.
A.In this condition, the repetition of letters is not allowed. This means, we cannot have the word in which both the letters are the same. Let us check how many choices each letter has in the two lettered words. In the first position, the total numbers of choices of letters are $={}^{4}{{C}_{1}}=4$ (as there are 4 letters in the word TIME). In the second position, there can be all letters except the letter present in the first position. So the total number of choices in second position are $={}^{3}{{C}_{1}}=3$ . Thus total number of words formed are $=4\times 3=12.$
B.In this condition, the repetition of letters is allowed. Let us check how many choices each letter has in the two lettered words. In the first position, the total number of choices are = 4. In the second position, the total number of choices are = 4 (because both the letters can be the same). Thus total number of two – lettered words formed are $=4\times 4=16.$
Hence, when the repetition of letter is not allowed, total words formed are = 12.
When the repetition of letters is allowed, total words formed are = 16.

Note: we can also solve the question by actually forming the words. In condition (a), the words formed = TI, TM, TE, IT, IM, IE, MT, MI, ME, ET, EI, EM. The total is 12. Similarly in condition (b), there will be 16 words.