Permutations and Combinations

A pizza parlor offers 10 toppings.

1. How many 3-topping pizzas could they put on their menu? Assume double toppings are not allowed.

2. How many total pizzas are possible, with between zero and ten toppings (but not double toppings) allowed?

3. The pizza parlor will list the 10 toppings in two equal-sized columns on their menu. How many ways can they arrange the toppings in the left column?

A combination lock consists of a dial with 40 numbers on it. To open the lock, you turn the dial to the right until you reach a first number, then to the left until you get to second number, then to the right again to the third number. The numbers must be distinct. How many different combinations are possible?

How many anagrams are there of the word “assesses” that start with the letter “a”?

On a business retreat, your company of 20 businessmen and businesswomen go golfing.

1. You need to divide up into foursomes (groups of 4 people): a first foursome, a second foursome, and so on. How many ways can you do this?

2. After all your hard work, you realize that in fact, you want each foursome to include one of the five Board members. How many ways can you do this?

Consider sets $A$ and $B$ with $|A| = 10$ and $|B| = 17$.

1. How many functions $f: A \to B$ are there?

2. How many functions $f: A \to B$ are injective?