The number of ways in which five identical balls can be distributed among ten identical boxes such that no box contains more than one ball is:

The number of $4$-letter words,with or without meaning,each consisting of $2$ vowels and $2$ consonants,which can be formed from the letters of the word $UNIVERSE$ without repetition is $.........$.

Let $S = \{1, 2, 3, \ldots, 9\}$. For $k = 1, 2, \ldots, 5$,let $N_k$ be the number of subsets of $S$,each containing five elements out of which exactly $k$ are odd. Then $N_1 + N_2 + N_3 + N_4 + N_5 =$

If ${ }^{11} C_4+{ }^{11} C_5+{ }^{12} C_6+{ }^{13} C_7={ }^{14} C_{r}$,then the value of $r$ is

The value of $\sum_{r=0}^4 {}^{(19-r)} C_3 + {}^{15} C_4$ is equal to

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then: