Answer to Question #345134 in Statistics and Probability for fitre

Question

Each time you visit the state fair, you play some games at the Midway. One game has four prizes (A, B, C, D). (a) How many different four letter sequences are possible if each letter is chosen from {A, B, C, D} with repeats allowed? (b) How many different four letter sequences have each letter occurring exactly once? (c) How many different four letter sequences don't contain B? (d) How many different four letter sequences have two letters, each of which occurs twice?

Answer 1

a) There are $4^4=256$ different four letter sequences are possible if each letter is chosen from {A, B, C, D} with repeats allowed.

b) The possible combinations, without using a letter more than once, are: $4!=24$ different 4-letter sequences.

c) Different four letter sequences that don't contain B are $n^k,$ where n = total number of elements in a set (4)

k = number of elements selected from the set (in this case - 3)

So, $4^3=64$

d) Different four letter sequences have two letters, each of which occurs twice are:

AABB, ABAB, BABA, BBAA, AACC, ACAC, CACA, CCAA, AADD, ADAD, DADA, DDAA, BBCC, BCBC, CBCB, CCBB, BBDD, BDBD, DBDB, DDBB, CCDD, CDCD, DCDC, DDCC.

So, 24.