4. Find the largest number of points which a foot ball team cannot get exactly using just 3-point field goals and 7-point touchdowns. Prove your answer is correct by mathematical induction.

Consider a popular ramen stall. Before the shop opens, there are 20people waiting in front of the restaurant.    Given that the shop only serves 10 bowls per day. If the maximum number ofeach menu the customers can order is 7 which means if the first 7 people orderShoyu Ramen, the rest must order Seafood Ramen. How many combinationof ramen orders(how each customer order and who got kicked out) are there?Ex: People can still order 5 Shoyu and 5 Seafood if they wish. But, no, the10 people who gets in can’t order 8 Shoyu and 2 Seafood.

There are 37 numbered balls in a drum (think lottery). A robot randomly picks 6 balls, I also randomly pick 6 balls. In how many ways can 4 of the balls picked by me be among the 6 balls picked by the robot?

If you have 12 boxes then how many balls do you have to put randomly into the boxes in order to guarantee that some box will contain at least 4 balls?

A teacher will randomly award three students with extra credit. How many possible ways can the teacher pick these three students from a class of 20 students?