Catherine rolls a $6$-sided die five times, and the product of her rolls is $600.$ How many different sequences of rolls could there have been? (The order of the rolls matters.)
There are 150 such sequences as follows:
14556 , 14565 , 14655 , 15456 , 15465 , 15546 , 15564 , 15645 , 15654 , 16455 , 16545 , 16554 , 22556 , 22565 , 22655 , 23455 , 23545 , 23554 , 24355 , 24535 , 24553 , 25256 , 25265 , 25345 , 25354 , 25435 , 25453 , 25526 , 25534 , 25543 , 25562 , 25625 , 25652 , 26255 , 26525 , 26552 , 32455 , 32545 , 32554 , 34255 , 34525 , 34552 , 35245 , 35254 , 35425 , 35452 , 35524 , 35542 , 41556 , 41565 , 41655 , 42355 , 42535 , 42553 , 43255 , 43525 , 43552 , 45156 , 45165 , 45235 , 45253 , 45325 , 45352 , 45516 , 45523 , 45532 , 45561 , 45615 , 45651 , 46155 , 46515 , 46551 , 51456 , 51465 , 51546 , 51564 , 51645 , 51654 , 52256 , 52265 , 52345 , 52354 , 52435 , 52453 , 52526 , 52534 , 52543 , 52562 , 52625 , 52652 , 53245 , 53254 , 53425 , 53452 , 53524 , 53542 , 54156 , 54165 , 54235 , 54253 , 54325 , 54352 , 54516 , 54523 , 54532 , 54561 , 54615 , 54651 , 55146 , 55164 , 55226 , 55234 , 55243 , 55262 , 55324 , 55342 , 55416 , 55423 , 55432 , 55461 , 55614 , 55622 , 55641 , 56145 , 56154 , 56225 , 56252 , 56415 , 56451 , 56514 , 56522 , 56541 , 61455 , 61545 , 61554 , 62255 , 62525 , 62552 , 64155 , 64515 , 64551 , 65145 , 65154 , 65225 , 65252 , 65415 , 65451 , 65514 , 65522 , 65541 , Total = 150 such sequences.
