Problem 2 Using concepts of Binomial Theorem and Combinations, find the coefficient of x50 in (1+x) + 2(1+x)> + 3(1+x)'+ ..+ 1000(1+x). Show the steps in your calculation. Problem 3 There is an easier way to implement the concept of Vigenere cipher on a message to obtain the ciphertext, using the concepts of modular arithmetic (that we have discussed in class) and/or performing simple arithmetic operations without the need to use the Vigenere square or the Vigenere table. Propose a step-by- step strategy of performing this. Your strategy should clearly state how the Vigenere square or the Vigenere table can be eliminated. Using your strategy obtain the ciphertext of the following message using the value of key = tpw Message = thereisasecretpassagebehindthepictureframe Note: no points will be provided if your solution involves computing the cipher text using the Vigenere square or the Vigenere table Problem 4 A detective figures that they have a one in nine chance of recovering a stolen property. Their out-of-pocket expenses for the investigation are $9,000. If they are paid their fee only if they recover the stolen property, what should they charge the client in order to break even? Problem 5 At Tucson Raceway Park, Tom's horse, H, has a probability of 1/20 of coming in first place, a probability of 1/10 of coming in second place, and a probability of 1/4 of coming in third place. First place pays $4,500 to the winner, second place $3,500 and third place $1,500. Is it worthwhile for Tom to enter the race if it costs $1,000? Problem 6 Do any of the following histograms represent a normal distribution? If yes, explain why by briefly discussing the concept of the Central Limit Theorem. Frequency Frequency 15 10 O 3 -2 -1 0 0.0 0.5 1.0 1.5 2.0