In a lottery, 5 balls are selected from a bin of twenty balls numbered 1 through 20. (The order doesn't matter.) Professor Luckless always guesses a 5-ball combination that includes her lucky number 7, and does not include her unlucky number 13. How many different days can she play the lottery without repeating a 5-ball guess?

Aspacecraft can attain a stable orbit 300 kilometers above earth if it reaches a velocity of 7.7 kilometers per second. the formula for a rocket's maximum velocity v in kilometers per second is vequalsminus0.0098tplusc ln upper r, where t is the firing time in seconds, c is the velocity of the exhaust in kilometers per second, and r is the ratio of the mass of the rocket filled with fuel to the mass of the rocket without fuel. find the velocity of a spacecraft whose booster rocket has a mass ratio of 20, an exhaust velocity of 2.1 km/s, and a firing time of 15 s. can the spacecraft achieve a stable orbit 300 km above earth?