Don't say "number" unless you are including decimals and fractions, positive or
negative. Either say "digit" or "single numeral". You can't have the number
-689.732 or 1/2 as a single character on a license plate, yet they are numbers.
So, I'll change your problem to this:
A certain state uses license plates that have six characters. The first two
characters can be either letters or DIGITs. The last four characters should be
consonant letters.
There are 10 digits, 0,1,2,3,4,5,6,7,8,9
There are 26 letters, A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z
There are 21 consonant letters: B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z
There are 5 vowels (non-consonant letters) A,E,I,O,U
How many license plates can be made if:
a. repetition is allowed?
Pick the 1st character as any one of these 36 characters:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z
Pick the second character as any one of these 36 characters.
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z
Pick the third character as any one of these 21 characters.
B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z
Pick the fourth character as any one of these 21 characters.
B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z
Pick the fifth character as any one of these 21 characters.
B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z
Pick the sixth character as any one of these 21 characters.
B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z
Answer: 36x36x21x21x21x21 - 362214 = 429981696
How many license plates can be made if:
b. repetition is not allowed?
Case 1: digit, digit,consonant,consonant,consonant,consonant.
(10)(9)(21)(20)(19)(18) = 12927600
Case 2: digit, vowel,consonant,consonant,consonant,consonant.
(10)(5)(21)(20)(19)(18) = 7182000
Case 4. digit, consonant,consonant,consonant,consonant,consonant.
(10)(21)(20)(19)(18)(17) = 24418800
Case 5: vowel,digit,consonant,consonant,consonant,consonant.
(5)(10)(21)(20)(19)(18) = 7182000
Case 6: vowel,vowel,consonant,consonant,consonant,consonant.
(5)(4)(21)(20)(19)(18) = 2872800
Case 7: vowel, consonant,consonant,consonant,consonant,consonant.
(5)(21)(20)(19)(18)(17) = 12209400
Case 8: consonant,digit,consonant,consonant,consonant,consonant.
(21)(10)(20)(19)(18)(17) = 24418800
Case 9: consonant,vowel,consonant,consonant,consonant,consonant.
(21)(5)(20)(19)(18)(17) = 12209400
Case 10: consonant,consonant,consonant,consonant,consonant,consonant.
(21)(20)(19)(18)(17)(16) = 39070080
You add 'em up.
Edwin
