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The digits 2,4,6,8 and 0 are used to make five digit numbers with no digit repeated. What is the probability that a number chosen at random for these numbers has the property that the digit in the hundreds place is larger than the digit in the tens place.

 Dec 11, 2020
 #1
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5! - 4! =96 permutations

20468 , 20486 , 20648 , 20684 , 20846 , 20864 , 24068 , 24086 , 24608 , 24680 , 24806 , 24860 , 26048 , 26084 , 26408 , 26480 , 26804 , 26840 , 28046 , 28064 , 28406 , 28460 , 28604 , 28640 , 40268 , 40286 , 40628 , 40682 , 40826 , 40862 , 42068 , 42086 , 42608 , 42680 , 42806 , 42860 , 46028 , 46082 , 46208 , 46280 , 46802 , 46820 , 48026 , 48062 , 48206 , 48260 , 48602 , 48620 , 60248 , 60284 , 60428 , 60482 , 60824 , 60842 , 62048 , 62084 , 62408 , 62480 , 62804 , 62840 , 64028 , 64082 , 64208 , 64280 , 64802 , 64820 , 68024 , 68042 , 68204 , 68240 , 68402 , 68420 , 80246 , 80264 , 80426 , 80462 , 80624 , 80642 , 82046 , 82064 , 82406 , 82460 , 82604 , 82640 , 84026 , 84062 , 84206 , 84260 , 84602 , 84620 , 86024 , 86042 , 86204 , 86240 , 86402 , 86420 , Total =  96 permutations.

 

20486 , 20684 , 20864 , 24086 , 24680 , 24860 , 26084 , 26480 , 26840 , 28064 , 28460 , 28640 , 40286 , 40682 , 40862 , 42086 , 42680 , 42860 , 46082 , 46280 , 46820 , 48062 , 48260 , 48620 , 60284 , 60482 , 60842 , 62084 , 62480 , 62840 , 64082 , 64280 , 64820 , 68042 , 68240 , 68420 , 80264 , 80462 , 80642 , 82064 , 82460 , 82640 , 84062 , 84260 , 84620 , 86042 , 86240 , 86420 , Total =  48 permutations.

 

Probability = 48 / 96 = 1 / 2

 Dec 11, 2020
 #2
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Sorry, I figured out the digits in the tens > digits in ones place. Here is the list where the digits in the hundreds > digits in the tens. The number DOES NOT change:

 

20648 , 20846 , 20864 , 24608 , 24806 , 24860 , 26408 , 26804 , 26840 , 28406 , 28604 , 28640 , 40628 , 40826 , 40862 , 42608 , 42806 , 42860 , 46208 , 46802 , 46820 , 48206 , 48602 , 48620 , 60428 , 60824 , 60842 , 62408 , 62804 , 62840 , 64208 , 64802 , 64820 , 68204 , 68402 , 68420 , 80426 , 80624 , 80642 , 82406 , 82604 , 82640 , 84206 , 84602 , 84620 , 86204 , 86402 , 86420 , Total =  48 permutations.

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