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## Homework Statement

Rachel invited her friends to dinner. She has 10 friends, but only 6 places to sit them in her (circular) table.

a) Count the ways to sit the guests if order is not important.

b) If order is important.

c) There is a married couple among her friends, and she want them to sit together. Count the ways to sit to sit the guests, if the order of the rest is not important.

d) If order is important.

e) Two of her friends are enemies, so she doesn't want them to sit together. Count the ways to sit the guests, if the order of the rest is not important.

f) If order is important.

g) Her friends are 4 women and 6 men. She wants to have at least 2 women sitted in the table. Count the ways to sit the guests if order is not important.

h) Count the ways to sit the guests if there are only men sitted at the table.

i) Count the ways to sit the guests if she chooses 3 men and 3 women and has them sitted interspersed.

## Homework Equations

Combinations: C(n, k) = n!/(k! * (n-k)!)

Permutations: P(n, k) = n!/(n-k)!

Permutations of n objects in a circle: (n - 1)!

## The Attempt at a Solution

a) C(10, 6) = 210

b) C(10, 6) * (6 - 1)! = 25200

c) 2! * C(8, 4) = 140

d) 2! * C(8, 4) * (5 - 1)! = 3360

e) 210 - 140 = 70 (a and d)

f) 25200 - 3360 = 21840 (b and d)

g) C(8, 4) = 70

h) C(6, 6) * (6 - 1)! = 120

i) 3! * C(6, 3) * 3! * C(4, 3) = 2880