How many 5letter words have exactly two vowels vowels are a

How many 5-letter words have exactly two vowels (vowels are \'a\', \'e\', i\', \'o\', and \'u\' again), OR have all their letters distinct, OR have their letters appearing in alphabetical order? For example, you should count \"abuzz\" (two vowels and in increasing order), \"alike\" (all letters distinct), \"aaaaa\" (letters in alphabetical order), but not things like \"issue\".

Solution

There are 5 vowels and 21 consonants,

a) Number of 5-letter words with exactly two vowels are (21)^3*(5)^2=231,525

b) Number of 5-letter words with all letters distinct is 26*25*24*23*22=7,893,600

c) Number of 5-letter words with letters appearing in alphabetical order is

26 if we count only letters \"aaaaa\", \"bbbbb\", ....,

25*(25+1)/2=325 if we count letters like aaaab, bbbbc, ...

24*(24+1)/2=300 if we count letters like aaabc, cccde, ...

23*(23+1)/2=276 if we count letters like aabcd, eefgh

22*(22+1)/2=253 if we count letters like abcde, efghi

The total number of ways is 26+325+300+276+253=1180