hat is the expected number of hat is the expected number of dates Bowser may have with Peach? Justify your answer carefully:

show your work and explain your calculation.

$($d$)(4$ point$)$ After a few dates, Bowser and Peach reached an agreement that if Bowser can solve the problem below,

then they will get married. Here is the problem. Every time, Bowser will pick a costume uniformly randomly

and exclusively from the first closet, until the closet is empty. If the key labelled on the costume is the smallest

compared to the ones he picked before, then Mario will take the costume away. The costume that is given to

Mario should be considered during the comparison as well. Note that, every costume in the first closet can only

be picked once. How many costumes is Mario expected to have? Justify your answer carefully: show your

work and explain your calculationdates Bowser may have with Peach? Justify your answer carefully:

show your work and explain your calculation.

$($d$)(4$ point$)$ After a few dates, Bowser and Peach reached an agreement that if Bowser can solve the problem below,

then they will get married. Here is the problem. Every time, Bowser will pick a costume uniformly randomly

and exclusively from the first closet, until the closet is empty. If the key labelled on the costume is the smallest

compared to the ones he picked before, then Mario will take the costume away. The costume that is given to

Mario should be considered during the comparison as well. Note that, every costume in the first closet can only

be picked once. How many costumes is Mario expected to have? Justify your answer carefully: show your

work and explain your calculation