Math: Permutations with identical items

[kingwinner]

Combinatorics is hard Can someone help me with these confusing questions. Thanks a lot!

 

1) Given 2 blue boxes, 1 red box, and 1 green box, how many different ways are there to arrange the 2 blue boxes if the blue boxes are indistinguishable?

 

Is the answer 1 or 2 for a question worded like this? I am not sure...

 

2) Eight football players are huddled in a circle. How many different arrangements of the players are possible? (Note: we have repetitive arrangements by rotating the members one place)

 

The answer is 8! / 8 = 5040

 

I don't understand why I should "divide by 8" intead of subtracting 8 from 8 factoral...What is the logic behind it?

 

3) How many different 6-letter arrangements can be formed from the word THIRST if T must always immediately precede H?

 

I can solve this problem if there are not identical letters, e.g. THIRSA gives (1x4x3x2x1)x5=120 arrangements. The problem here is that two are 2 "T"s in the word THIRST. How can I cope with it?

 

4) How many three-letter "words" can be formed using the letters of SLEEP?

 

5) A necklace is composed of 7 beads and a clasp. How many different necklaces can be made if: (HINT: consider if you FLIP a necklace, some arrangements are repeated)

a) 6 beads are of the same size and 1 bead is larger

:lol: The beads are of two sizes, 5 of the smaller size and 2 of the larger size

For part a), my first try is (7! / 6!) x (1/2), I multiply by (1/2) because if you flip the necklace, you get some repeated arrangements...but it seems wrong as I think deeper, if the larger bead is in the middle (opposite to the clasp), the arrangement is the same even if you flip it...I am so confused! (Does part b also have the same problem?)