If $2 \times {}^nC_5 = 9 \times {}^{n-2}C_5$,then the value of $n$ is:

There are $10$ persons named $A, B, \dots, J$. We have space to keep only $5$ of them. If $A$ must be included,and $G$ and $H$ must not be included in the group of $5$,in how many ways can we arrange the group in a row?

Out of $7$ consonants and $4$ vowels,the number of words consisting of $3$ consonants and $2$ vowels that can be formed is:

$A$ bag contains $5$ red marbles,$4$ black marbles,and $3$ white marbles. The number of ways in which $4$ marbles can be drawn so that at most $2$ of them are red is:

$A$ lady gives a dinner party for $6$ guests. The number of ways in which they may be selected from among $10$ friends,if two of the friends will not attend the party together,is:

If $^nC_3 + ^nC_4 > ^{n+1}C_3$,then