If 4 letter words are formed using the letters A B C D E F G

If 4 -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: No condition is imposed. The letter C must be at the end and letters can be repeated

Solution

there are 7 letters with two vowels.

a) if no conditions are imposed then it can be

the firt letter can be any one of 7 letters, the second one can also be any one from 7 letters, similar argument for the third and hence for the fourth

hence total number of combinations is =7*7*7*7=2401 [answer]

b) no letters can be repeated= all 4 are different=⁷C₄*4!=840 [from 7 select 4 letters,they can be arranged in 4! ways] [answer]

c) the first letter is A and repeatation is allowed.

hence the second one can be any one from 7 letters,similar argument for the third and hence for the fourth

hence answer is 7*7*7=343 [answer]

d) the last letter is A and repeatation is allowed.

hence the first one can be any one from 7 letters,similar argument for the second and hence for the third

hence answer is 7*7*7=343 [answer]

e) the second letter must be a vowel and letters can be repeated.

then the first can be anyone from 7 letters,so the third and so the fourth

but the second can be any one from A or E.so only two choices

hence total combinations=7*2*7*7=686 [answer]