Let A be the set of 26 letters of the alphabet, in lowercase. Let S be the set o

Question

Let A be the set of 26 letters of the alphabet, in lowercase. Let S be the set of six-long letter strings, in which letters may repeat. Find the size of each of the following subsets. (Your answer can be a number, or a product. You may use nCk for "n choose k") (A) S itself. (B) The subset B Subsetequalto S of all strings in which no letter appears more than once. (C) The subset C in which the letters a, e, i, o, u do not appear. (D) The subset D in which the fourth and last letters agree. (The same letter may appear elsewhere.) (E) The subset E in which only the fourth and last letters agree. That is, no other letters in two different positions agree. (F) The subset F all strings that contain exactly three copies of the letter x.

Explanation / Answer

a) as for each letter we have 26 choices hence number of letters =266 =308915776

b)as each letter can appear once ; therefore number of letters =26C6 =230230

c)as we need 6 letters out of remaining 21 letters therefore number of letters =216 =85766121

d)as 4th and last letter agree ; we need to select 5 letters out of 26 with no repetition ; therefore ; number of ways =26C5 =65780

e)as there are 3 copies of letter x; therefore remaing 3 letters each has 25 choices ; hence number of letters=253

=15625

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