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The number of ways in which $5$ girls and $3$ boys can be seated in a row so that no two boys are together is:

In how many ways can $8$ people stand in a line such that there are always exactly two people between two specific people $A$ and $B$?

There are $9$ balls to be placed in $9$ boxes. If $5$ of the balls cannot fit into $3$ specific small boxes,find the number of ways to arrange exactly one ball in each of the boxes.

If ${}^9P_5 + 5 \cdot {}^9P_4 = {}^{10}P_r$, then the value of $r$ is:

The number of ways to form a $7$-digit number using the digits $1, 2, 3, 4, 3, 2, 1$ such that odd digits always occupy odd places is:

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