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)?

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 format

C1 - C2 - C3 - C4 - C5

C1 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

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... So

26-25-24-23-22

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)?

Start with a, end with any letter other than z.

1-26-26-26-25

End with z, start with any other letter than A

25-26-26-26-1

Start with A, end with Z

1-26-26-26-1

896,376 of these words start with an a or end with a z or both.

entering that problem on the tube might get you an answer

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional