(9 points) Jared is asked in class to decode a message by reordering letters in a certain order, but unfortunately, they never learned how to read. Jared can only distinguish what each letter looks like, and is trying every possible ordering of letters, since they cannot read. Thus far, they only know the letters of the first part of the message: aeeegginnnoopruuvvy There are 19 total characters in the first part of the message. (a) (3 points) In how many ways can Jared order the letters (without spaces)? (Note that repeated letters are considered identical. For example, the first ' e ' in the string is indistinguishable from the other two 'e' characters in the string) (b) (3 points) Taking pity on Jared, the instructor, Dr Astley, gives them the full words in the message to arrange to the correct message, and lets them know that 'never' is always immediately followed by 'gonna'. Jared, still unable to read, attempts every valid ordering possible with this newfound information. In total, there are 28 words, and 19 distinct words, in this message. In how many ways can Jared order these words? (repeated words, like 'you' are considered the same as the other instances of 'you' in the string) Below is a table of each of the counts of words, followed by a list of words that have that count in the message: (c) (3 points) Next, Prof Astley wants to arrange the words in the message using the words from part c) into a circle, while rotating the message around, so that the message scrolls around the circle indefinitely. In doing so, because of the rotation, some messages are considered indistinct from others. For example if the message was "never gonna give you up" arranged in a circle clockwise in that order, this is considered indistinguishable from the message "up never gonna give you" since the first message can be rotated around to match the second message. With the same words and counts as part c), and with "never" always immediately followed by "gonna", in how many ways can Jared arrange these words in a circle