Question 5 (Combinations and \'stars and bars\') A TAW is a three letter \'word

Question

Question 5 (Combinations and 'stars and bars') A TAW is a three letter 'word whose letters are in alphabetical order. Letters are drawn from the standard lower case English 26-letter alphabet, and are allowed to repeat. A word' does not have to appear in any dictionary but must contain at least one vowel and at least one consonant. Letter y can count as a vowel or a consonant. Examples of TAWs are abc, ccy, aoy and yyy. How many different TAWs are there? Hint 1: Once three letters are chosen (possibly involving repeats) there is only one way to put them in alphabetical order Hint 2: As a first step ignore the requirement about vowels and consonants

Explanation / Answer

Firstly, if you really want to combine them without them wanting to make sense,please refer the answers already provided, and as they say, with repetition it is 26x26x26 and without repetition it is 26x25x24.

Mathematically, it is possible to combine 3 out of 26 alphabets in this way-

Repetition - Choose 1 out of 26 for each letter, i.e. 3 times. It will give you 26C1 x 26 C1 x 26C1 = 26x26x26 = 17576.

Without repetition - Choose 1 out of 26 for first letter, 1 out of remaining 25 for second, 1 out of the rest of 24 for the third. You get 26C1 x 25C1 x 24C1 = 26x25x24 = 15600.

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