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. The subset B S of all strings in which no letter appears more than once. The subset C in which the letters a, e, i, o, u do not appear. The subset D in which the fourth and last letters agree. (The same letter may appear elsewhere.) The subset E in which only the fourth and last letters agree. That is, no other letters in two different positions agree. The subset F all strings that contain exactly three copies of the letter x.

Explanation / Answer

1) number of ways 6letters can be chosen out of 26 where each letter repeats itself =266 as for every letter we have 26 choices.

2)as no letter appears twice permutaion of 6 letters out of 40 =(26)6=26*25*24*23*22*21=1.66*108

C)without a.e.i.o.u we have 21 choices

hence number of ways =216

d)as 4th and last letter agree we need to arrange only 5 letters, hence number of ways =265

e)for fourth and last letter we have 26 choice,

and for other four to not to repeat we have 25*24*23*21 ways, hence total number of way =26*25*24*23*21=7534800

f)for 3 copies of X, rest 3 letter can be chose in 253 ways in b/w then 3 x can be arrange in 10 ways

hence total number of ways =253*10=156250

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