The alphabet contains 26 letters, 5 of them are vowel. How many codes consisting
of 5 letters, 3 consonants and 2 vowels can be formed?
It doesn't say whether letters can be repeated. I'll assume they
cannot be repeated. [If letters can be repeated tell me in the thank-
you note form below and I'll get back to you with that solution.]
From this amount, how many are there that
a) contain the letter b?
There are 26-5= 21 consonants, including b.
Choose "b" 1C1 way.
There are 20 consonants besides "b".
Choose the 2 consonants besides "b" in 20C2 ways.
Choose the 2 vowels 5C2 ways.
The 5 letters can be arranged in 5! ways.
That's 1C1*20C2*5C2*5! = 1*190*10*120 = 228000
b) contain both the letters b and c?
There are 26-5 consonants, including "b" and "c".
Choose "b" 1C1 way.
Choose "c" 1C1 way,
There are 19 consonants besides "b" and "c".
Choose the 1 consonant besides "b" and "c" in 19C1 ways.
Choose the 2 vowels 5C2 ways.
The 5 letters can be arranged in 5! ways.
That's 1C1*19C1*5C2*5! = 1*19*10*120 = 22800
c) contain the letter b and end with c?
There are 26-5 consonants, including "b" and "c".
Choose "b" 1C1 way.
Choose "c" 1C1 way to place at the end.
There are 19 consonants besides "b" and "c".
Choose the 1 consonant besides "b" and "c" in 19C1 ways.
Choose the 2 vowels 5C2 ways.
The 4 letters other than the "c" at the end can be
arranged in 4! ways.
That's 1C1*19C1*5C2*4! = 1*19*10*24 = 4560
Edwin