Question

Six-letter code sequences are formed from the letters A through Z (26 letters). The letters can be repeated.
(a) What is the probability that a six-letter sequence formed starts with X, Y, or Z?

(b) What is the probability that 2 X's, 2 Y's, and 2 Z's form the six-letter sequence?

Answer 1

(a) Probability that a six letter sequence formed starts with X,Y or Z ?

Each letter can be selected 26 ways ( A,B...Z).

So, total ways 6 letters can be selected = 26 * 26 * 26 * 26 * 26 * 26 = 308915776

six letter sequence can be formed which starts with X,Y or Z = 3 * 26 * 26 * 26 * 26 * 26=35644128

Probability that a six letter sequence formed starts with X,Y or Z =  35644128 / 308915776 = 0.1154

Hence, Probability that a six letter sequence formed starts with X,Y or Z is 0.1154 or 11.54%.

(b) What is the probability that 2 X's, 2 Y's, and 2 Z's form the six-letter sequence?

Total ways first 2 similar letters , say X  can be placed on 6 positions =  

Total ways next 2 similar letters, say Y can be placed on remaining 4 position =

Total ways next 2 similar letters, say Z can be placed on remaining 2 position =

So, ways 2 X, 2Y and 2Z can form the six letter sequence =  

total ways 6 letters can be selected = 26 * 26 * 26 * 26 * 26 * 26 = 308915776

probability that 2 X's, 2 Y's, and 2 Z's form the six-letter sequence = ways 2 X, 2Y and 2Z can form the six letter sequence / total ways 6 letters can be selected  

probability that 2 X's, 2 Y's, and 2 Z's form the six-letter sequence =

Hence, probability that 2 X's, 2 Y's, and 2 Z's form the six-letter sequence is 0.0000002913