Question

A person has to create an identification tag for their team. The identification tag of their team must contain exactly 7 letters. The first letter must be either a J, N, or R. The second letter must be a vowel that is not an E or U. The third letter must be a F, G, H, or M. The fourth letter must be a consonant that is not a B, D, or L. The fifth letter must be a consonant. The sixth letter must be a vowel. If repetition of letters is not allowed, how many different identification tags can the person make for their team? (Note: a vowel is A, E, I, O, or U, and consonants are letters that are not vowels). Show your work to receive credit

Answer 1

Vowel : A, E, I , O, U

Consonants : B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z- Number consonants = 26-5 =21

repetition of letters is not allowed

The First letter must either a J,N,R - Number of ways - 3

The second letter must be a vowel that is not an E or U - i.e either (A,I,O) - Number of ways - 3

The third letter must be a F, G, H, or M - Number of ways - 4

The fourth letter must be a consonant that is not a B, D, or L

2 consonant Already used for( First letter - J or N or R) and third letter (F or G or H or M) and Consonants B, D, L can not used : Consonants left :21-2-3=16 - Number of ways - 16

The fifth letter must be a consonant - One more Consonant is used for fourth letter- Number of Consonants left : 21-3 =18 - Number of ways - 18

The sixth letter must be a vowel - Number of vowels left - 4 (as A or I or O used for second letter) - Number of ways - 4

The Seventh letter can be any of the consonants or the vowels left : Consonant left : 18-1=17;

Vowels left : 4-1 =3 ; Number of ways = 17+3 =20

Number of different identification tags can the person make for their team = 3 x 3 x 4 x 16 x 18 x 4 x 20 = 829440

Number of different identification tags can the person make for their team = 829440