MAT103 PROBLEM SET 2
Directions: Please write your solutions neatly on a long bondpaper and take a clear photo and submit
though MOLE classroom on or before 5pm of December 15, 2020.
1. Prove that 1
21+2
22+3
23+ โฏ + ๐
2๐< 2 for all ๐ โฅ 1.
2. Your wardrobe consists of 5 shirts, 3 pairs of pants and 17 bow ties. How many different outfits
can you make?
3. A pizza parlor offers 10 toppings.
a. How many 3-topping pizzas could they put on their menu? Assume double toppings are not
allowed.
b. How many total pizzas are possible, with between zero and ten toppings (but not double
toppings) allowed?
c. 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?
4. How many different seating arrangements are possible for King Arthur and his 9 knights around
their round table?
5. Consider the identity:
๐(๐
๐) = ๐ (๐ โ 1
๐ โ 1)
a. Is this true? Try it for a few values of ๐ and ๐.
b. Use the formula for (๐
๐) to give an algebraic proof of the identity.
c. Give a combinatorial proof of the identity.
