Answer following counting questions. How many permutations of the seven letters

Question

Answer following counting questions. How many permutations of the seven letters A, B,C,D,E, F,G are there? a) low many permutations of the seven letters A, B, C, D, E, F, G have the two vowels before the five consonants? b) Find the number of subsets of S = { 1, 2, 3, … , 10) that contain exactly five elements, all of them even. In how many ways can you get a committee of 4 freshmen and 3 sophomores when you have a class with 30 students 10 freshmen, 12 sophomores, and 8 juniors. Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin with L or end with R. d) e)

Explanation / Answer

Solution:

a)

consider it like we have 7 boxes in which we will be placing these letters

Number of permutations= 7! = 7P7 ways= 5040

b)

Two vowels before five consonants can be placed in two ways

AE_ _ _ _ _

and

EA_ _ _ _ _

Rest of them can be placed in 5! ways

which means 2*5!

c)

Out of 10 we have to choose 5

they can be chosen in

10C5 ways which will be calculated to

252, and to make it only for even numbers

it will be

252/2 = 126

d)

10C4 X 12 C 3 = 46200

e)

So we have 8 boxes here

from which 2 are filled

L _ _ _ _ _ _ R

Rest of the 6 can be filled in 26^6 ways since repetition is allowed

So 26^6 ways.

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