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Three dice are rolled.  Find the probability that all the rolls are different, and none of them are equal to six.

 May 9, 2020
 #1
avatar+903 
-2

6 outcomes/ dice, so 6^3 = 216 total outcomes

 

There are 6 ways of getting a trio of rolls with same numbers, 111,222,...666

 

We also have the ones with 2 numbers the same. 112,113, etc.

 

Each of the repeating digit(there's 6) has 3 positions to put the other non-same number, front, center, last, and 5 numbers to choose from, so 6*5*3 = 90 ways.

 

We ALSO have the ones with 3 different rolls but equal to 6. Each of these has 3! = 6 ways of arrangements.

 

1+2+3

 

I honestly don't think there are any else.

 

SO that's 6.

 

Adding all the ways, 6+6+90 = 102

 

So I'm pretty sure the answer's 102/216 = 51/108 and you can simplify that

 

If you don't understand feel free to ask.

 May 9, 2020
 #2
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When you say "None of them is equal to six", do you mean the sum of each permutation, or 6 itself must not appear in any permutation?

 May 9, 2020
edited by Guest  May 9, 2020
edited by Guest  May 9, 2020
 #3
avatar+903 
-2

I assume the question meant none of the rolls could have a sum of 6

hugomimihu  May 9, 2020
 #4
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+1

If that is what he/she meant, then I get a different result as follows:

 

124 , 125 , 126 , 134 , 135 , 136 , 142 , 143 , 145 , 146 , 152 , 153 , 154 , 156 , 162 , 163 , 164 , 165 , 214 , 215 , 216 , 234 , 235 , 236 , 241 , 243 , 245 , 246 , 251 , 253 , 254 , 256 , 261 , 263 , 264 , 265 , 314 , 315 , 316 , 324 , 325 , 326 , 341 , 342 , 345 , 346 , 351 , 352 , 354 , 356 , 361 , 362 , 364 , 365 , 412 , 413 , 415 , 416 , 421 , 423 , 425 , 426 , 431 , 432 , 435 , 436 , 451 , 452 , 453 , 456 , 461 , 462 , 463 , 465 , 512 , 513 , 514 , 516 , 521 , 523 , 524 , 526 , 531 , 532 , 534 , 536 , 541 , 542 , 543 , 546 , 561 , 562 , 563 , 564 , 612 , 613 , 614 , 615 , 621 , 623 , 624 , 625 , 631 , 632 , 634 , 635 , 641 , 642 , 643 , 645 , 651 , 652 , 653 , 654 , Total =  114

 

 

So, the probability is: 114 / 6^3 =19 /36

 May 9, 2020
 #5
avatar+903 
-5

No... I might not have been clear.

 

what I meant to say was that it could be that the three rolls have a total sum of 6, and your answers include sums of more than 6.

hugomimihu  May 10, 2020

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