Question 1204819
If 5 -letter words are formed using the letters {{{A}}}, {{{B}}}, {{{C}}}, {{{D}}}, {{{E}}}, {{{F}}}, {{{G}}}, how many such words are possible for each of the following conditions:


(a) No condition is imposed.

There are {{{7}}} choices for the first letter.
There are  {{{7}}} choices for the second letter
There are  {{{7}}} choices for the third letter.
There are  {{{7}}} choices for the fourth letter
There are  {{{7}}} choices for the fifth letter.

The total number of {{{5}}}-letter words possible:

{{{7* 7 * 7 *7*7 = 16807}}}

Your answer is : {{{16807}}}


(b) No letter can be repeated in a word.

For the first letter, there are {{{7}}} choices.
For the second letter, there are only {{{6}}}choices left (since we cannot repeat the first letter). 
For the third letter there are {{{5}}} choices. 
For the fourth letter there are  {{{4}}} choices. 
For the fifth letter there are {{{3 }}} choices.

The total number of {{{5}}}-letter words possible with no letter repeated:

{{{7 * 6* 5 * 4 *3= 2520}}}

Your answer is : {{{2520}}}



(c) Each word must begin with the letter {{{A}}} and letters can be repeated.

For the first letter, there is only {{{1}}} choice (the letter {{{A}}}). 
For the second letter there are {{{7}}} choices (since any letter can be used).
For the third letter there are {{{7 }}}choices,  for the fourth letter there are {{{7 }}}choices, and for the fifth letter there are {{{7}}} choices.

The total number of {{{5}}}-letter words is possible with the first letter being {{{A}}}:

{{{1 * 7 * 7 *7 *7= 2401}}}

Your answer is : {{{2401}}}


(d) The second letter must be a vowel and no letters can be repeated.

For the first letter, there are {{{7}}} choices (since any letter can be used). 
For the second letter, there are {{{2}}} choices (the vowels {{{A}}}, and {{{E}}}).
For the third letter there are {{{5}}} choices, for the fourth letter there are {{{4}}} choices, and for the fifth letter there are {{{34}}} choices.

The total number of {{{5}}}-letter words possible with the second letter being a vowel:

{{{7 *2 *5 *4*3 = 840}}}

Your answer is : {{{840}}}