In how many ways can $8$ identical balls be distributed into $3$ distinct boxes such that no box remains empty?

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ containing $20$ distinct elements is $k$ times the number of $5$-element subsets containing the element $a_4$,then what is the value of $k$?

If ${}^{10}P_r = 604800$ and ${}^{10}C_r = 120$,then $r =$

There are $6$ different novels and $3$ different poetry books on a table. If $4$ novels and $1$ poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle,then the number of such possible arrangements is

If a person has $3$ coins of different denominations,the number of different sums that can be formed is

There are $15$ people at a party and each person shakes hands with every other person. What is the total number of handshakes?