1 - How mny ways are there to have a collectio of eight fruits from a large pile of identical oranges, apples, bananas, peaches and pears if the collection should nclude exactly two different kinds of fruits? 2 - How many numbers greater than 3,000,000 can be formed by arrangements of 1, 2, 2, 4 , 6, 6 ,6? 3 - How many 8-digit seuences are there invloving exactly six different digits? 4 - How many arrangments are possible with five letters chosen from MSSISSIPPI? 5 - How many arrangements of letters in REPETITION are there with the first E occuring before the first T? 6 - how many arrangements of the letters in MATHEMATICS are there in which TH appear together but the TH is not immediately followed by an E (not THE)? Please answer whatever you can. and show WORK so I can understand how you got the answer. I will give the points to the one who does it correctly and show his/her work of doing the problem Solution 6. You are dealing with permutations here, because you are trying to count different arrangements of the letters subject to two constraints (the arrangement must contain the letters TH in that order but cannot contain the letters THE in that order). If you didn\'t have any constraint, then you have to calculate all the arrangements of 11 letters, which is 11! (11 factorial). But in the letters there are two \"A\"s, two \"M\"s and two \"T\"s, and swapping the two \"A\"s (say) around in any arrangement simply gives the same arrangement. So you have to divide 11! by 2 x 2 x 2 = 8 to give the total number of arrangements of MATHEMATICS without any constraints. How to deal with the constraints? I suggest that you think of \"TH\" as a single letter. This then means that the total number of arrangements drops to 10!/8 (again you have to adjust for the two \"A\"s and the two \"M\"s, and also for the two ways in which you can get \"TH\", given the two \"T\"s). Then on the same argument the total number of arrangements containing the sequence \"THE\" is 9!/8. So the number of arrangements that satisfy both constraints is: 10!/8 - 9!/8 = 9! x 9/8 (as 10! = 9! x 10) This is 408,240..