Q:

Consider all 5 letter "words" made from the full English alphabet. (a) How many of these words are there total? (b) How many of these words contain no repeated letters? (c) How many of these words start with an a or end with a z or both (repeated letters are allowed)?

Accepted Solution

A:
Answer:a) There are 11,881,376 of these words in total.b) 7,893,600 of these words contain no repeated letters.c) 896,376 of these words start with an a or end with a z or both.Step-by-step explanation:The English alphabet has 26 letters.Our word has the following formatC1 - C2 - C3 - C4 - C5C1 is the first character, C2 the second, etc...(a) How many of these words are there total?Each of C1, C2,... can be 26. So the total is[tex]T = (26)^{5} = 11,881,376[/tex]There are 11,881,376 of these words in total.(b) How many of these words contain no repeated letters?C1 can be any letter. C2 can be any letter bar the letter at C1. C3 any other than C2, C1... So26-25-24-23-22[tex]T = 26*25*24*23*22 = 7,893,600[/tex]7,893,600 of these words contain no repeated letters.(c) How many of these words start with an a or end with a z or both (repeated letters are allowed)?[tex]T = T_{1} + T_{2} + T_{3}[/tex][tex]T_{1}:[/tex] Start with a, end with any letter other than z.1-26-26-26-25[tex]T_{1} = 26^{3}*25 = 439,400[/tex][tex]T_{2}:[/tex]End with z, start with any other letter than A25-26-26-26-1[tex]T_{2} = 26^{3}*25 = 439,400[/tex][tex]T_{3}:[/tex]Start with A, end with Z1-26-26-26-1[tex]T_{3} = 17,576[/tex][tex]T = T_{1} + T_{2} + T_{3} = 439,400 + 439,400 + 17,576 = 896,376[/tex]896,376 of these words start with an a or end with a z or both.