A group of $n$ boys can be arranged in a line in a certain number of ways. By adding two more boys to the group, the number of possible arrangements increases by a factor of $420.$ Find the value of $n.$

Find the number of distinct arrangements of all the letters in these words:

$\mathrm{TABLE}$

Find the number of distinct arrangements of all the letters in these words:

TABLET

Find the number of distinct arrangements of all the letters in these words:

COMMITTEE

Find the number of distinct arrangements of all the letters in these words:

MISSISSIPPI

Find the number of distinct arrangements of all the letters in these words:

HULLABALLOO.

Find how many six-digit numbers can be made from these sets of digits:

$1,1,1,1,1$ and $3$

Find how many six-digit numbers can be made from these sets of digits:

$2,2,2,7,7$ and $7$

Find how many six-digit numbers can be made from these sets of digits:

$5,6,6,6,7$ and $7$