Question

How many different ways are possible to arrange the letters of the word MACHINE so that the vowels may occupy only the odd positions?

Answer 1

notice all letters in \(MACHINE\) are unique. let's write out a possible "blank slate":

Answer 2

a,i e occupies position 1,3,5,7 so 4p3=12ways

Answer 3

the 1st, 3rd, 5th, and 7th positions will be occupied by the consonants
while the 2nd, 4th, and 6th will be occupied by the vowels

Answer 4

only vowels occupy odd positions

Answer 5

oh oops I misread!

Answer 6

you suck haha

Answer 7

4!*3! total

Answer 8

24*6=144 ways

Answer 9

you suck more hahaha

Answer 10

actually \(_4P_1=4!/(4-1)!=4!/3!=4\) i.e. \(4\) ways to pick the \(3\) odd spots for vowels...

anyways now we just fill in the others. we count the consonants separately (just fill in the even spots and the one unoccupied odd spot) so \(4!\)... there are also \(3!\) ways to assign the vowels to the \(3\) spots we picked so our answer is just \(4\cdot4!\cdot3!=576\)

Answer 11

how

Answer 12

4 how

Answer 13

yeah