The value of ${ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4$ is

$^nC_r + ^{n-1}C_r + ...... + ^rC_r =$

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 $k$?

All the five-digit numbers in which each successive digit exceeds its predecessor are arranged in increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

$A$ group of $9$ students,$s_1, s_2, \ldots, s_9$,is to be divided into three teams $X, Y$,and $Z$ of sizes $2, 3$,and $4$,respectively. Suppose that $s_1$ cannot be selected for team $X$,and $s_2$ cannot be selected for team $Y$. The number of ways to form such teams is:

In a library,there are $n$ distinct books and each has $p$ copies. What is the number of ways to select one or more books?