Lottery.$$Your dreams of becoming rich have just come true. You have won the State of Tranquility's Lottery. The State offers you two payment plans for the $$5,000,000$ advertised jackpot. You can take annual payments of $$250,000$ at the end of the year for the next $20$ years or $$2,454,537$ today.

a$.$ If your investment rate over the next $20$ years is $10\%,$ which payoff will you choose?

b$.$ If your investment rate over the next $20$ years is $7\%,$ which payoff will you choose?

c$.$ At what investment rate will the annuity stream of $$250,000$ be the same as the lump$-$sum payment of $$2,454,537?$