Consider relation Nobel$($name$,$category,year$),$ recording winner names, categories, and years of Nobel prizes. The category could be "peace", or "chemistry", for example. Consider the following SQL expression:

with N as

$($select name, count$(*)$ as number

from Nobel

group by name$)$

select $*$

from N natural join

$($select max$($number$)$ as number from N$)$

Which of the following statements are correct?

$1.$

An equivalent way to write the same query in SQL is as follows:

with N as

$($select name, count$(*)$ as number

from Nobel

group by name$)$

select $*$

from $($select name from Nobel$)$ natural join

$($select max$($number$)$ as number from N$)$

$2.$

This expression returns the names of people who won the most Nobel prizes, together with how many prizes that was.

$3.$

This expression returns the names of people who won Nobel prizes in the most different categories, together with how many categories that was.

$4.$

An equivalent way to write the same query in SQL is as follows:

select name, max$($number$)$

from $($select name, count$(*)$ as number

from Nobel

group by name$)$

group by name

$5.$

This expression returns, for each Nobel prize winner, the number of prizes they have won.