1) equation has 5 vowels and 3 consonants V = vowel 1V2V3V4V5V6as you can see, there are 6 places we could put the consonants such that none of them touch each other. There are 3 different consonants, we want to figure out how many different ways we can place them in those 3 slots (worded differently, take 3 out of those 6 slots), and order does matter, so we want to use the probability function nPr. n = 6, r = 3, so 6P3. To take into account the different ways in which the vowels can be arranged (your teacher was nice and gave you a word with no repeats), we multiply 6P3 by 5! (factorial). Each of the 5 vowels could be in the first spot, there are 4 left over, each of which could fill the next spot, etc. 5*4*3*2*1. I don't have a calculator on hand so you'll have to plug it in yourself. 6P3 * 5!2) a)This problem is similar to the first, but doesn't require a calculator. Like before,B = boy 1B2B3B4B5B6So we have 6 places to put 5 girls, or more simply, 6 places to not put a girl. Answer = 6b) if you didn't put a girl in spots 2-5, then two boys would be next to each other. This means we can only NOT put a girl in spots 1 and 6. Answer = 2hope I didn't mess up

enjoyedit: amy, the consonants can be in more than one position.edit again: oh, yea, listen to the guy below me