Arrangements with repetition and restrictions

If some items are identical, swapping them does not make a new arrangement. Divide by the factorial of each repeat:

$\frac{n!}{a!b!\dots }$

Example. The letters of BANANA (6 letters, with A three times and N twice):

$\frac{6!}{3!2!}=60$

The question will say whether repeated letters are treated as identical or different. Read it carefully.

Kept together: treat the group as one block. Arrange the blocks, then arrange the items inside the block.

Example. 4 boys and 3 girls stand in a row with the girls together. The girls form one block, so there are 5 units:

$5!\times 3!=720$

Kept apart: use the complement.

$\text{apart}=\text{total}-\text{together}$

Example. 10 different books with 2 particular books not next to each other:

$10!-9!\times 2!=2903040$

Place the fixed items first, then arrange the rest.

Example. The letters of GEOMETRY (8 letters, E twice). How many arrangements start with G and end with Y?

G and Y are fixed, leaving 6 letters with E twice:

$\frac{6!}{2!}=360$

Out of $\frac{8!}{2!}=20160$ arrangements, the probability is $\frac{360}{20160}=\frac{1}{56}$.